Orthogonal Vectors Problem (OV)

• Orthogonal Vectors Problem (OV) is a Boolean decision problem that determines whether two binary vectors have an inner product of zero, making it a central benchmark in fine-grained complexity.
• The problem underpins equivalence results across diverse domains, linking data structure lower bounds, approximation algorithms, and reductions via communication protocols and locality-sensitive hashing.
• OV drives research on algorithmic barriers in low and moderate dimensions, influencing conjectures like the Orthogonal Vectors Conjecture and advancing techniques in circuit complexity and average-case analysis.

Searching arXiv for Orthogonal Vectors and related recent work. Searching arXiv for exact paper (Chen et al., 2018). The Orthogonal Vectors problem, usually abbreviated $\textsf{OV}$, is a Boolean-vector decision problem that asks whether a given collection contains a pair of vectors with inner product zero. In its standard red–blue formulation, the input consists of two sets $A,B \subseteq \{0,1\}^d$ of size $n$, and the task is to decide whether there exists $(a,b)\in A\times B$ such that $a\cdot b=0$ (Chen et al., 2018). In the low-dimensional regime $d=c\log n$, $\textsf{OV}$ is a central benchmark in fine-grained complexity because a naive algorithm runs in $O(n^2 d)$, hence $O(n^2\log n)$, while the Orthogonal Vectors Conjecture posits that no truly subquadratic algorithm exists in this regime (Chen et al., 2018). The problem serves simultaneously as a canonical hardness source, a hub for equivalence results, a target for data-structure lower bounds, and a testbed for average-case, circuit, and low-dimensional algorithmic techniques (Chen et al., 2018, Kane et al., 2017).

1. Formal definition and parameter regimes

The monochromatic formulation of $\textsf{OV}$ asks, given $A,B \subseteq \{0,1\}^d$0, whether there exist $A,B \subseteq \{0,1\}^d$1 such that

$A,B \subseteq \{0,1\}^d$2

Equivalently, two vectors are orthogonal when they have no coordinate where both entries are $A,B \subseteq \{0,1\}^d$3 (Chen et al., 2018). The red–blue formulation, used extensively in reductions, takes two sets $A,B \subseteq \{0,1\}^d$4 with $A,B \subseteq \{0,1\}^d$5 and asks whether some $A,B \subseteq \{0,1\}^d$6 and $A,B \subseteq \{0,1\}^d$7 satisfy $A,B \subseteq \{0,1\}^d$8 (Chen et al., 2018).

The literature distinguishes several dimensional regimes. The low-dimensional or sparse regime is $A,B \subseteq \{0,1\}^d$9 for arbitrary constant $n$0, or more generally $n$1 in geometric variants (Chen et al., 2018). A moderate-dimensional regime takes $n$2 for constant $n$3 (Chen et al., 2018, Abboud et al., 2018). These regimes matter because both algorithms and conjectured lower bounds are highly sensitive to $n$4. The naive algorithm checks all $n$5 pairs and computes each inner product in $n$6, giving $n$7 time, which becomes $n$8 for $n$9 (Chen et al., 2018).

A standard fine-grained notion is “truly subquadratic time”: a problem is solvable in truly subquadratic time if there exists $(a,b)\in A\times B$0 such that for every constant $(a,b)\in A\times B$1, instances with $(a,b)\in A\times B$2 can be solved in $(a,b)\in A\times B$3 time (Chen et al., 2018). In geometric settings, the analogous notion is a $(a,b)\in A\times B$4-approximation in $(a,b)\in A\times B$5 time when dimension and coordinate bit-length are $(a,b)\in A\times B$6 (Chen et al., 2018).

The $(a,b)\in A\times B$7-OV generalization takes $(a,b)\in A\times B$8 sets $(a,b)\in A\times B$9 of $a\cdot b=0$0 binary vectors and asks whether there is a tuple $a\cdot b=0$1 whose coordinatewise product is zero in every coordinate (Dalirrooyfard et al., 27 Mar 2025, Dürr et al., 15 Jul 2025). This higher-arity variant is important in average-case hardness and low-dimensional algorithms, but the classical fine-grained literature centers on the $a\cdot b=0$2 case.

2. Orthogonal Vectors Conjecture and its fine-grained role

The Orthogonal Vectors Conjecture, or OVC, is commonly stated as follows: for some $a\cdot b=0$3 and all sufficiently large $a\cdot b=0$4, there is no algorithm solving $a\cdot b=0$5 on $a\cdot b=0$6 vectors in dimension $a\cdot b=0$7 in time $a\cdot b=0$8 (Chen et al., 2018). A broader formulation says that for $a\cdot b=0$9, $d=c\log n$0 requires $d=c\log n$1 time (Kane et al., 2017).

OVC is implied by the Strong Exponential Time Hypothesis. The standard chain is

$d=c\log n$2

with reductions due to Williams and later work (Chen et al., 2018, Abboud et al., 2018). In moderate dimensions, the conjecture states that there are no $d=c\log n$3 such that $d=c\log n$4 with $d=c\log n$5 can be solved in $d=c\log n$6 time (Abboud et al., 2018).

Within fine-grained complexity, $d=c\log n$7 functions as a hardness core for quadratic-time phenomena. Many conditional lower bounds for problems in strings, graphs, computational geometry, formal languages, and dynamic algorithms reduce from $d=c\log n$8 (Abboud et al., 2018). Before large equivalence results were known, $d=c\log n$9 was mainly used in one direction: if a target problem admitted a truly subquadratic algorithm, then OVC would fail. Subsequent work showed that in low dimensions, many apparently harder inner-product and geometric problems are in fact equivalent to $\textsf{OV}$0 under truly-subquadratic reductions (Chen et al., 2018).

The importance of $\textsf{OV}$1 is not limited to worst-case running time. The problem also appears in data structures, where the online query version becomes equivalent to Partial Match and related tasks (Chen et al., 2018, Goldstein et al., 2017, Gajulapalli et al., 6 May 2026). It appears in circuit complexity as an explicit function with near-optimal branching-program and formula lower bounds (Kane et al., 2017). It also appears in average-case fine-grained hardness through parity variants and planted distributions (Dalirrooyfard et al., 27 Mar 2025, Kühnemann et al., 30 Apr 2025).

3. Equivalence classes around OV

A central development is the identification of a broad equivalence class around low-dimensional $\textsf{OV}$2. The paper "An Equivalence Class for Orthogonal Vectors" shows that, in dimension $\textsf{OV}$3, the following problems are either all solvable in truly subquadratic time or none are: $\textsf{OV}$4, $\textsf{OV}$5, $\textsf{OV}$6, $\textsf{OV}$7, constant-factor approximate $\textsf{OV}$8 and $\textsf{OV}$9, additive approximate $O(n^2 d)$0, additive approximate Jaccard-Index-Pair, approximate bichromatic $O(n^2 d)$1-Closest Pair for constant $O(n^2 d)$2, and approximate $O(n^2 d)$3-Furthest Pair in dimension $O(n^2 d)$4 (Chen et al., 2018).

The Boolean inner-product problems are defined on red–blue sets $O(n^2 d)$5. For example,

$O(n^2 d)$6

and

$O(n^2 d)$7

(Chen et al., 2018). Since $O(n^2 d)$8 is equivalent to deciding whether $O(n^2 d)$9, reductions among these problems are immediate in one direction. The nontrivial direction is showing that $O(n^2\log n)$0, $O(n^2\log n)$1, and approximation variants reduce back to $O(n^2\log n)$2 in the same low-dimensional regime (Chen et al., 2018).

Two reduction frameworks underpin these equivalences. The first uses efficient $O(n^2\log n)$3 communication protocols. If a Boolean predicate $O(n^2\log n)$4 has a small $O(n^2\log n)$5 communication protocol, then the corresponding satisfying-pair problem can be reduced to a small number of $O(n^2\log n)$6 instances, with only $O(n^2\log n)$7 blowup and dimension preserved at $O(n^2\log n)$8 (Chen et al., 2018). Inner-product equality admits such a protocol, yielding $O(n^2\log n)$9, and from there the rest of the Boolean IP family follows (Chen et al., 2018).

The second framework uses locality-sensitive hashing. If a metric or similarity predicate admits a $\textsf{OV}$0-sensitive LSH family with gap $\textsf{OV}$1, then the corresponding satisfying-pair problem reduces to additive $\textsf{OV}$2 approximation to $\textsf{OV}$3 in dimension $\textsf{OV}$4 (Chen et al., 2018). Since $\textsf{OV}$5 metrics for $\textsf{OV}$6 and Jaccard index admit LSH families, approximate closest pair, furthest pair, and Jaccard pair become equivalent to additive $\textsf{OV}$7, hence to $\textsf{OV}$8 (Chen et al., 2018).

The same paper also establishes a data-structure equivalence: there is a $\textsf{OV}$9-space, $A,B \subseteq \{0,1\}^d$00-query-time data structure for Partial Match on $A,B \subseteq \{0,1\}^d$01 if and only if such a data structure exists for $A,B \subseteq \{0,1\}^d$02-approximate nearest neighbor search in $A,B \subseteq \{0,1\}^d$03, $A,B \subseteq \{0,1\}^d$04 (Chen et al., 2018). This situates $A,B \subseteq \{0,1\}^d$05 at the center of a larger online/offline equivalence landscape.

In moderate dimensions, the equivalence picture becomes more fragmented. The same work shows that moderate-dimensional $A,B \subseteq \{0,1\}^d$06 is truly-subquadratically equivalent to moderate-dimensional approximate $A,B \subseteq \{0,1\}^d$07, while $A,B \subseteq \{0,1\}^d$08, $A,B \subseteq \{0,1\}^d$09, and $A,B \subseteq \{0,1\}^d$10 form another equivalence class (Chen et al., 2018). Whether these moderate-dimensional classes fully merge remains open in that paper.

4. Algorithms and parameterized improvements

The classical brute-force baseline for $A,B \subseteq \{0,1\}^d$11 is $A,B \subseteq \{0,1\}^d$12, or $A,B \subseteq \{0,1\}^d$13 in low dimension (Chen et al., 2018). For small $A,B \subseteq \{0,1\}^d$14, folklore algorithms also exploit enumeration over all $A,B \subseteq \{0,1\}^d$15 subsets, giving $A,B \subseteq \{0,1\}^d$16 time in the set formulation (Dürr et al., 15 Jul 2025). A significant line of work aims to reduce the $A,B \subseteq \{0,1\}^d$17 dependence in low dimension and to improve the exponent on $A,B \subseteq \{0,1\}^d$18 in the $A,B \subseteq \{0,1\}^d$19 regime.

For low-dimensional $A,B \subseteq \{0,1\}^d$20-OV, Williams gave a $A,B \subseteq \{0,1\}^d$21-time algorithm via a succinct equality-rank decomposition of the disjointness matrix (Dürr et al., 15 Jul 2025). A later combinatorial algorithm improves this to randomized $A,B \subseteq \{0,1\}^d$22, and with computer-aided parameter optimization to $A,B \subseteq \{0,1\}^d$23 (Dürr et al., 15 Jul 2025). The algorithm uses a representation-method style certificate search. The universe $A,B \subseteq \{0,1\}^d$24 is partitioned into blocks, random families of subsets are sampled in each block, and a tuple of sampled subsets serves as a certificate if one input vector is contained blockwise while another is disjoint blockwise. Entropy estimates bound the number of certificates that need to be enumerated, producing the $A,B \subseteq \{0,1\}^d$25 and $A,B \subseteq \{0,1\}^d$26 bases (Dürr et al., 15 Jul 2025).

The same paper generalizes this to $A,B \subseteq \{0,1\}^d$27-OV. For every fixed $A,B \subseteq \{0,1\}^d$28, there exists $A,B \subseteq \{0,1\}^d$29 such that $A,B \subseteq \{0,1\}^d$30-OV can be solved in time

$A,B \subseteq \{0,1\}^d$31

(Dürr et al., 15 Jul 2025). The algorithm combines two ideas. First, a down-closure inclusion–exclusion method counts orthogonal $A,B \subseteq \{0,1\}^d$32-tuples in time proportional to the total size of the down-closures of the input set families. Second, if some sets are large, the algorithm guesses them and recurses on lower-arity OV over the remaining intersection. A conditional lower bound based on the Set Cover Conjecture shows that for any $A,B \subseteq \{0,1\}^d$33, there exists $A,B \subseteq \{0,1\}^d$34 such that a $A,B \subseteq \{0,1\}^d$35-time algorithm for $A,B \subseteq \{0,1\}^d$36-OV would contradict Set Cover, suggesting that $A,B \subseteq \{0,1\}^d$37 as $A,B \subseteq \{0,1\}^d$38 grows is unavoidable (Dürr et al., 15 Jul 2025).

For average-case OV, a 2024 paper studies inputs where each coordinate is independently $A,B \subseteq \{0,1\}^d$39 with probability $A,B \subseteq \{0,1\}^d$40 and gives a new algorithm running in

$A,B \subseteq \{0,1\}^d$41

for dimension $A,B \subseteq \{0,1\}^d$42, for any $A,B \subseteq \{0,1\}^d$43 (Alman et al., 2024). This improves on the prior worst-case exponent $A,B \subseteq \{0,1\}^d$44 of Abboud–Williams–Yu in the hardest parameter regime where $A,B \subseteq \{0,1\}^d$45 may depend on $A,B \subseteq \{0,1\}^d$46 (Alman et al., 2024). The key idea is to use the simple real polynomial

$A,B \subseteq \{0,1\}^d$47

and sum its evaluations over blocks of vectors. Orthogonal pairs contribute $A,B \subseteq \{0,1\}^d$48, while typical random pairs have inner product concentrated around $A,B \subseteq \{0,1\}^d$49, making their total contribution small with high probability (Alman et al., 2024). The analysis tracks how polynomial values degrade as inner products move away from orthogonality.

In another direction, "Kronecker Powers, Orthogonal Vectors, and the Asymptotic Spectrum" gives deterministic algorithms for moderate-dimensional $A,B \subseteq \{0,1\}^d$50 based on depth-2 circuits for the disjointness matrix $A,B \subseteq \{0,1\}^d$51, where

$A,B \subseteq \{0,1\}^d$52

(Alman et al., 17 Sep 2025). The paper shows that decision OV can be solved in deterministic time $A,B \subseteq \{0,1\}^d$53, derandomizing an earlier algorithm of Nederlof and Węgrzycki, and counting OV in $A,B \subseteq \{0,1\}^d$54, improving Williams’ $A,B \subseteq \{0,1\}^d$55 counting algorithm (Alman et al., 17 Sep 2025). The central framework reduces sparse vector–matrix–vector multiplication with $A,B \subseteq \{0,1\}^d$56 to evaluating low-degree depth-2 circuits for Kronecker powers. For decision OV, OR-circuits suffice and give asymptotic degree $A,B \subseteq \{0,1\}^d$57; for counting, partition circuits give asymptotic degree $A,B \subseteq \{0,1\}^d$58 (Alman et al., 17 Sep 2025).

The same circuit perspective also yields modular variants $A,B \subseteq \{0,1\}^d$59, and ties improvements in disjointness circuit degree directly to improved OV algorithms (Alman et al., 17 Sep 2025). This suggests a close interaction between low-dimensional OV algorithms and nontrivial depth-2 circuit constructions.

A specialized but notable algorithmic connection appears in the Subset Sum literature. "Improving Schroeppel and Shamir's Algorithm for Subset Sum via Orthogonal Vectors" reduces worst-case Subset Sum to a structured OV instance where vectors have fixed Hamming weight, notably support size $A,B \subseteq \{0,1\}^d$60, and gives an OV algorithm running in time

$A,B \subseteq \{0,1\}^d$61

for such instances (Nederlof et al., 2020). This reduction yields a Subset Sum algorithm with $A,B \subseteq \{0,1\}^d$62 time and $A,B \subseteq \{0,1\}^d$63 space, the first improvement over Schroeppel–Shamir’s longstanding space bound (Nederlof et al., 2020). A plausible implication is that even restricted fixed-weight OV can encode barriers for classical exponential-time algorithms.

5. Data structures and the online variant

The online or indexing version of OV asks for preprocessing a set $A,B \subseteq \{0,1\}^d$64 of $A,B \subseteq \{0,1\}^d$65 vectors so that a query vector $A,B \subseteq \{0,1\}^d$66 can be tested quickly for orthogonality against some stored vector (Goldstein et al., 2017, Gajulapalli et al., 6 May 2026). In fine-grained terms, this is $A,B \subseteq \{0,1\}^d$67. Earlier work on "Orthogonal Vectors Indexing" focused on the regime $A,B \subseteq \{0,1\}^d$68 and studied space–query tradeoffs (Goldstein et al., 2017).

That paper gives several explicit data structures. The simplest divide-by-ones strategy partitions vectors into those with few ones and those with many ones, using arrays indexed by coordinate blocks and bitmap acceleration for heavy buckets (Goldstein et al., 2017). More sophisticated query-graph constructions yield truly sublinear query time with space about $A,B \subseteq \{0,1\}^d$69 in the $A,B \subseteq \{0,1\}^d$70 regime (Goldstein et al., 2017). It also shows a general tradeoff: for any $A,B \subseteq \{0,1\}^d$71, one can achieve query time $A,B \subseteq \{0,1\}^d$72 with space $A,B \subseteq \{0,1\}^d$73 for some $A,B \subseteq \{0,1\}^d$74 (Goldstein et al., 2017). On random input vectors, the same paper improves the space to $A,B \subseteq \{0,1\}^d$75 for any $A,B \subseteq \{0,1\}^d$76, since bucket sizes in the query graph concentrate to $A,B \subseteq \{0,1\}^d$77 (Goldstein et al., 2017).

A 2026 revisit of OnlineOV substantially refines this picture (Gajulapalli et al., 6 May 2026). In low dimensions $A,B \subseteq \{0,1\}^d$78, it gives a deterministic data structure with almost-linear space $A,B \subseteq \{0,1\}^d$79 and query time $A,B \subseteq \{0,1\}^d$80, matching the best known randomized low-dimensional performance up to constant factors in the exponent while removing randomness (Gajulapalli et al., 6 May 2026). In moderate dimensions $A,B \subseteq \{0,1\}^d$81, it improves the older Charikar–Indyk–Panigrahy bounds, obtaining query time $A,B \subseteq \{0,1\}^d$82 and space

$A,B \subseteq \{0,1\}^d$83

(Gajulapalli et al., 6 May 2026). For $A,B \subseteq \{0,1\}^d$84, this gives sublinear query time and subexponential-in-$A,B \subseteq \{0,1\}^d$85 space, constituting the first deterministic refutation of a hardness conjecture proposed by Goldstein, Lewenstein, and Porat for $A,B \subseteq \{0,1\}^d$86 (Gajulapalli et al., 6 May 2026).

The main technical tool in that paper is a structure-versus-randomness decomposition. The input set is partitioned into a pseudorandom part, where every projection onto $A,B \subseteq \{0,1\}^d$87 coordinates yields only small candidate lists, and structured blocks where many vectors share zeros on a fixed coordinate set, permitting a recursive dimension reduction (Gajulapalli et al., 6 May 2026). This decomposition extends to Partial Match, Orthogonal Range Search, DNF Evaluation, and related online problems via standard reductions (Gajulapalli et al., 6 May 2026).

On the lower-bound side, under the Non-Uniform Strong Exponential Time Hypothesis, the same paper proves that for any constants $A,B \subseteq \{0,1\}^d$88 and $A,B \subseteq \{0,1\}^d$89, there exists a constant sparsity $A,B \subseteq \{0,1\}^d$90 such that no batch data structure with polynomial space $A,B \subseteq \{0,1\}^d$91 and amortized query time $A,B \subseteq \{0,1\}^d$92 exists for $A,B \subseteq \{0,1\}^d$93-Sparse-$A,B \subseteq \{0,1\}^d$94 with $A,B \subseteq \{0,1\}^d$95, even with computationally unbounded preprocessing (Gajulapalli et al., 6 May 2026). Through reductions, these lower bounds extend to Partial Match, Subset Query, DNF Evaluation, Orthogonal Range Search, and approximate nearest neighbors in polylogarithmic dimension (Gajulapalli et al., 6 May 2026). This gives an unusually strong conditional separation between preprocessing space and query complexity in online OV-like tasks.

6. Average-case, planted, and parity versions

The classical OVC is worst-case. Several recent works investigate whether $A,B \subseteq \{0,1\}^d$96 remains hard on natural or planted distributions.

One line concerns parity counting. "Average-Case Hardness of Parity Problems: Orthogonal Vectors, $A,B \subseteq \{0,1\}^d$97-SUM and More" studies parity-$A,B \subseteq \{0,1\}^d$98-OV, which asks for the parity of the number of orthogonal $A,B \subseteq \{0,1\}^d$99-tuples (Dalirrooyfard et al., 27 Mar 2025). The paper gives explicit distributions under which parity-$n$00-OV is average-case hard. In particular, for constant $n$01, it proves that parity-$n$02-OV in dimension $n$03 requires at least $n$04 time on average under rETH, and $n$05 time on average under $n$06-XOR or $n$07-SUM hypotheses (Dalirrooyfard et al., 27 Mar 2025). The method proceeds through factored versions of $n$08-OV, low-degree polynomial representations, worst-case-to-average-case reductions, and a factored-to-unfactored gadget that maps factored OV instances to standard $n$09-OV instances (Dalirrooyfard et al., 27 Mar 2025).

A different average-case direction studies planted instances. "The Planted Orthogonal Vectors Problem" defines a planted distribution for $n$10-OV where a unique orthogonal $n$11-tuple is embedded among $n$12-biased random vectors in dimension $n$13 with $n$14 (Kühnemann et al., 30 Apr 2025). The planted coordinate distribution is engineered so that any subset of fewer than $n$15 planted vectors has exactly the same marginal distribution as in the unplanted model, while the full $n$16-tuple is orthogonal (Kühnemann et al., 30 Apr 2025). This $n$17-wise indistinguishability allows average-case search-to-decision reductions: a fast decision algorithm for distinguishing planted from unplanted instances yields a fast algorithm for recovering the planted tuple (Kühnemann et al., 30 Apr 2025). The paper conjectures that these planted instances still require $n$18 time on average (Kühnemann et al., 30 Apr 2025).

The average-case landscape is subtle because some random distributions actually make OV easy. Kane and Williams showed that for every fixed $n$19, i.i.d. Bernoulli-$n$20 OV instances can be solved by $n$21 formulas of size $n$22 on all but an $n$23 fraction of instances, demonstrating that standard independent random restrictions do not capture worst-case hardness (Kane et al., 2017). The 2024 average-case algorithm of Alman, Andoni, and Zhang then strengthens the algorithmic side, showing that even in the hardest $n$24-dependent regime, random OV is easier than worst-case OV (Alman et al., 2024). This suggests that average-case hardness for OV, if it holds, likely depends on carefully planted or structured distributions rather than simple product distributions.

7. Circuit, communication, and broader complexity connections

Beyond algorithms and reductions, $n$25 has become a proxy for understanding the limits of current lower-bound techniques. The paper "The Orthogonal Vectors Conjecture for Branching Programs and Formulas" proves that in several nonuniform models, OVC is effectively true unconditionally (Kane et al., 2017). Specifically, OV on $n$26 vectors in dimension $n$27 has branching-program complexity, constant fan-in formula complexity, and symmetric-gate formula wire complexity

$n$28

up to polylogarithmic factors (Kane et al., 2017). In the regime $n$29 or larger, this yields essentially quadratic lower bounds, matching the strongest known explicit lower bounds for any function in those models (Kane et al., 2017). The proof uses carefully structured restrictions rather than independent random restrictions, and the same framework extends to Batch Partial Match and related OV-hard problems (Kane et al., 2017).

The paper also shows a limitation of common lower-bound methods: under i.i.d. Bernoulli-$n$30 input distributions, OV becomes easy for small-depth formulas, so independent random restrictions cannot prove worst-case hardness here (Kane et al., 2017). This resonates with later average-case algorithmic results and suggests that any robust unconditional progress on general OV lower bounds must use more structured arguments.

Communication complexity appears prominently in the equivalence-class work (Chen et al., 2018). Efficient $n$31 protocols for predicates such as exact inner product equality yield reductions to OV, turning communication upper bounds into fine-grained equivalences (Chen et al., 2018). Conversely, LSH transforms metric proximity problems into approximate $n$32, and hence to OV (Chen et al., 2018). This duality between communication protocols and fine-grained reductions has become one of the standard conceptual tools around OV.

Circuit complexity also reappears in the 2025 asymptotic-spectrum paper. There, the decision and counting OV algorithms arise from depth-2 linear circuits for Kronecker powers of the disjointness matrix, and OVC implies that the Walsh–Hadamard matrix and the disjointness matrix cannot have $n$33-size depth-2 circuits of certain types (Alman et al., 17 Sep 2025). Proving such circuit lower bounds would also imply breakthroughs for threshold circuits, indicating that stronger lower bounds in this direction are difficult for reasons already familiar from classical circuit complexity (Alman et al., 17 Sep 2025).

Finally, OV connects to quantum complexity. "Fine-grained quantum supremacy based on Orthogonal Vectors, 3-SUM and All-Pairs Shortest Paths" encodes low-dimensional OV into the acceptance probability of specific quantum circuits (Hayakawa et al., 2019). Under OV or gap-OV conjectures, exact or multiplicative-error classical simulation of those circuits is impossible in certain exponential times as a function of the number of qubits $n$34 (Hayakawa et al., 2019). In the QRAM model, the circuits have linear size in $n$35, and the lower bound on classical simulation time is

$n$36

for suitable $n$37 from the OV conjecture (Hayakawa et al., 2019). This places OV alongside 3-SUM and APSP as a polynomial-time fine-grained assumption capable of supporting quantum-supremacy style claims.

A related consequence appears in "More Consequences of Falsifying SETH and the Orthogonal Vectors Conjecture" (Abboud et al., 2018). That paper shows that if the moderate-dimensional OV conjecture fails, then for all $n$38 and large enough $n$39, weighted $n$40-Clique and Min-Weight-$n$41-Clique on $n$42-hypergraphs admit $n$43-time algorithms, and sparse threshold-circuit SAT obtains unexpected sub-$n$44 algorithms (Abboud et al., 2018). It also proves that the Weighted Clique conjecture implies the moderate-dimensional OV conjecture (Abboud et al., 2018). This situates OV inside a larger hierarchy of conjectures tied to weighted clique, APSP, and sparse circuit satisfiability.

Orthogonal Vectors therefore occupies a distinctive position. It is simple enough to encode as disjointness over subsets, strong enough to underpin dozens of fine-grained lower bounds, flexible enough to admit equivalence results across combinatorial, geometric, and data-structure problems, and rich enough to support advances in average-case complexity, circuit lower bounds, and exponential-time algorithm design (Chen et al., 2018, Kane et al., 2017, Alman et al., 17 Sep 2025). The persistent tension between its deceptively elementary definition and the breadth of its implications is the reason it remains one of the central benchmark problems in modern fine-grained complexity.