Compiled Nonlocal Games Overview

• Compiled nonlocal games are constructions that re-express Bell-type nonlocality as game-theoretic frameworks or single-prover protocols using utility functions or cryptographic protocols.
• They leverage quantum homomorphic encryption to transform multi-prover interactions into sequential protocols while reliably simulating spatial separation and preserving quantum correlations.
• Operator-algebraic and semidefinite methods are applied to rigorously certify soundness, ensuring both quantum completeness and classical security with negligible error.

Compiled nonlocal games are constructions that re-express Bell-type nonlocality in a different operational format. One strand of the literature compiles a Bell scenario into a Bayesian game whose utilities encode Bell-type correlations, so that nonlocality appears as a game-theoretic resource rather than only as a violation of a win–lose Bell inequality (Pappa et al., 2014). A second, now central, strand compiles a multi-prover non-local game into a single-prover interactive protocol by using quantum homomorphic encryption (QHE) to hide some questions and thereby simulate spatial separation cryptographically (Kalai et al., 2022). In that cryptographic setting, the subject comprises the compiler itself, the preservation of classical and quantum values, the operator-algebraic and semidefinite techniques used to analyze compiled protocols, and their applications to verification and self-testing (Kulpe et al., 2024).

1. Terminological scope and early Bell-to-game compilation

In the Bell-theoretic sense, compilation means embedding the correlation structure of a Bell scenario into a utility function. “Nonlocality and conflicting interest games” presents a two-player Bayesian game and explicitly describes it as a direct bridge between Bell nonlocality, Bayesian game theory, and what is often called a compiled nonlocal game viewpoint: a Bell scenario is “compiled” into a game whose payoffs encode the Bell-type correlations the players can generate (Pappa et al., 2014). The novelty there is that the game is not a common-interest game such as CHSH. For three of the four input pairs the players are rewarded for coordination, while for the input pair $x_A=x_B=1$ they are rewarded for anti-coordination; the utilities are asymmetric, so the Bell-derived game has conflicting interests rather than a shared win–lose objective. The classical payoff region satisfies a Bell-type inequality $F_A+F_B\le 9/8$, while a CHSH-style quantum strategy using $|\phi^+\rangle$ yields $F_A=F_B=\frac{3}{4}\cos^2(\pi/8)\approx 0.64$, strictly above the best fair classical equilibrium $F_A=F_B=9/16$ (Pappa et al., 2014).

The cryptographic literature uses compilation differently. There, compilation does not change the objective into a new payoff function; instead it preserves the original non-local game predicate while changing the interaction pattern. The verifier still samples the original questions, but sends some of them encrypted to a single prover, later reveals the remaining question in the clear, decrypts the early answer, and checks the original verification predicate (Kalai et al., 2022). This later notion is the one around which the modern theory of compiled nonlocal games is organized.

2. The KLVY compiler and the single-prover protocol model

In the cryptographic formulation, a $k$-player non-local game is specified by a PPT-sampleable query distribution $Q$ over tuples $(q_1,\dots,q_k)\in (\{0,1\}^n)^k$ and a polynomial-time verification predicate $V(q_1,\dots,q_k,a_1,\dots,a_k)\in\{0,1\}$, where each answer $a_i\in\{0,1\}^m$ (Kalai et al., 2022). The classical value is the maximum success probability over local deterministic answer maps $F_A+F_B\le 9/8$0, and the entangled value is the maximum over a shared state and local unitaries applied to $F_A+F_B\le 9/8$1 followed by measurement (Kalai et al., 2022).

The KLVY compiler of Kalai et al. turns such a game into a single-prover, sequential interactive protocol. In the two-prover case, the verifier samples $F_A+F_B\le 9/8$2, generates a QHE key, encrypts the first question, sends the ciphertext, receives an encrypted first answer, then sends the second question in the clear and receives the second answer; acceptance is defined by decrypting the first answer and evaluating the original predicate (Kalai et al., 2022). For $F_A+F_B\le 9/8$3 provers, the verifier encrypts the first $F_A+F_B\le 9/8$4 questions under independent keys, receives encrypted answers one by one, sends the final question in the clear, receives the final answer, decrypts, and checks the original predicate. This produces a $F_A+F_B\le 9/8$5-round single-prover protocol (Kalai et al., 2022).

The key cryptographic requirement is QHE with an auxiliary-input correctness property. Correctness is required not only on a standalone plaintext but also in the presence of entanglement with an auxiliary register, so that homomorphic evaluation preserves the joint correlations with the untouched part of the state (Kalai et al., 2022). This matters because the honest quantum strategy is assumed to be efficiently implementable: the compiler must homomorphically evaluate the first $F_A+F_B\le 9/8$6 prover circuits while preserving the entanglement needed for the later cleartext round (Kalai et al., 2022).

3. Completeness, classical soundness, and the baseline guarantees

The original KLVY theorem establishes the two baseline properties of the compiler. For any $F_A+F_B\le 9/8$7-prover non-local game with quantum value $F_A+F_B\le 9/8$8 and classical value $F_A+F_B\le 9/8$9, and any QHE scheme satisfying auxiliary-input correctness and able to homomorphically evaluate at least $|\phi^+\rangle$0 prover strategies, there is a $|\phi^+\rangle$1-round single-prover interactive game with completeness $|\phi^+\rangle$2 against a QPT prover and soundness at most $|\phi^+\rangle$3 against any PPT classical prover (Kalai et al., 2022). In that sense, the compiler preserves quantum completeness and classical soundness up to negligible additive loss.

The soundness proof is constructive. In the two-player case, a deterministic cheating classical prover for the compiled game is converted into a pair of local provers for the original game: one prover emulates the tail of the cheating strategy on the cleartext question, and the other hardwires the first-round transcript and chooses the first answer maximizing conditional acceptance probability (Kalai et al., 2022). The proof then uses a semantic-security reduction, together with an approximate maximization argument based on sampling and a Chernoff bound, to show that if the compiled protocol were noticeably easier to cheat than the original game, encryptions of different questions could be distinguished (Kalai et al., 2022). For $|\phi^+\rangle$4, the argument becomes a hybrid proof across the encrypted rounds (Kalai et al., 2022).

The compiler also admits amplification. For a constant-size game $|\phi^+\rangle$5 with $|\phi^+\rangle$6, sequential repetition or random-terminating parallel repetition can yield a repeated game with quantum value $|\phi^+\rangle$7 and classical value $|\phi^+\rangle$8; under a threshold parallel repetition hypothesis of the form $|\phi^+\rangle$9, polynomially secure QHE suffices for the compiled protocol to achieve quantum value $F_A=F_B=\frac{3}{4}\cos^2(\pi/8)\approx 0.64$0 and classical value $F_A=F_B=\frac{3}{4}\cos^2(\pi/8)\approx 0.64$1 (Kalai et al., 2022). CHSH is the canonical example: compiling CHSH with Mahadev’s QFHE gives a four-round single-prover game with quantum computational-soundness value at least $F_A=F_B=\frac{3}{4}\cos^2(\pi/8)\approx 0.64$2 and classical computational-soundness value at most $F_A=F_B=\frac{3}{4}\cos^2(\pi/8)\approx 0.64$3 (Kalai et al., 2022).

4. Quantum soundness: from CHSH to general bipartite and multipartite games

The principal question left open by the baseline compiler was quantum soundness: whether an efficient quantum prover in the compiled game can exceed the intended quantum benchmark of the original game. The first sharp result was game-specific. Natarajan and Zhang proved that the compiled CHSH game still satisfies the Tsirelson bound, with compiled quantum value $F_A=F_B=\frac{3}{4}\cos^2(\pi/8)\approx 0.64$4, and they recovered a strong rigidity statement in which Bob’s two observables approximately anticommute on the relevant post-measurement state (Natarajan et al., 2023).

This was extended from CHSH to broader classes. For any two-player XOR game, the compiled value obeys

$F_A=F_B=\frac{3}{4}\cos^2(\pi/8)\approx 0.64$5

and, together with KLVY completeness, this yields $F_A=F_B=\frac{3}{4}\cos^2(\pi/8)\approx 0.64$6 asymptotically; the same work also proves tight compiled parallel repetition for XOR games and operator self-testing statements (Cui et al., 2024). A parallel development showed that the compilation procedure preserves the quantum bound not only for XOR games but also for $F_A=F_B=\frac{3}{4}\cos^2(\pi/8)\approx 0.64$7-outcome CHSH/SATWAP-type games, and used the resulting compiled games for computational self-testing of arbitrary pairs of qubit measurements and of the elegant Bell inequality’s Pauli triple (Baroni et al., 2024).

The general bipartite asymptotic theorem replaced these game-specific analyses with a universal bound. For every two-player nonlocal game $F_A=F_B=\frac{3}{4}\cos^2(\pi/8)\approx 0.64$8 and every QPT strategy $F_A=F_B=\frac{3}{4}\cos^2(\pi/8)\approx 0.64$9,

$F_A=F_B=9/16$0

so the correct universal upper bound is the commuting-operator value of the underlying game, not merely its ordinary finite-dimensional quantum value (Kulpe et al., 2024). The paper also situates this theorem within the standard correlation hierarchy

$F_A=F_B=9/16$1

with the corresponding value chain $F_A=F_B=9/16$2 (Kulpe et al., 2024). The multipartite analogue was then proved for all finite $F_A=F_B=9/16$3-player games: asymptotically, any compiled QPT strategy yields a value at most $F_A=F_B=9/16$4 (Baroni et al., 16 Jul 2025).

Quantitative soundness came later. For bipartite compiled Bell games whose optimal quantum strategy is finite-dimensional, any polynomial-time prover’s score is negligibly close to the game’s ideal quantum value; for arbitrary bipartite games, the compiled score is upper-bounded by a newly formalized sequential NPA hierarchy plus a negligible term (Klep et al., 22 Jul 2025). A subsequent result established quantitative quantum soundness for all multipartite compiled nonlocal games: for every level $F_A=F_B=9/16$5, every efficient prover strategy is upper-bounded by the level-$F_A=F_B=9/16$6 value of a multipartite sequential NPA hierarchy plus a negligible term, and if the original game has a finite-dimensional optimal strategy then the compiled score is at most $F_A=F_B=9/16$7 (Baroni et al., 29 Sep 2025).

A recurrent misconception is that compilation automatically preserves the ordinary tensor-product quantum value for every game. The general theorems are more nuanced. Universally, the asymptotic upper bound is $F_A=F_B=9/16$8, and for games without finite-dimensional optimal strategies the finite-level sequential-NPA approximation error can be essential; the literature explicitly relates this issue to the conjecture $F_A=F_B=9/16$9 and to the absence of a universal computable convergence rate for NPA across all games (Kulpe et al., 2024, Klep et al., 22 Jul 2025).

5. Operator-algebraic and semidefinite frameworks

The core technical development behind general soundness is the reformulation of compiled protocols as sequential nonlocal strategies. In the bipartite case, a compiled QPT strategy at security parameter $k$0 can be represented as

$k$1

where $k$2 are unnormalized post-measurement states after the first encrypted round and $k$3 is Bob’s second-round POVM (Kulpe et al., 2024). Such strategies are automatically non-signaling from Bob to Alice but may signal from Alice to Bob. The decisive strengthening is a strong non-signaling condition: independence of the total Bob-side state from Alice’s question must hold not only for Bob’s raw POVM statistics but for all noncommutative polynomials in Bob’s POVM elements (Kulpe et al., 2024). In finite dimensions, a strongly non-signaling sequential strategy induces a quantum correlation in $k$4; in general infinite dimensions, the correct target is $k$5 (Kulpe et al., 2024).

The proof architecture is operator-algebraic. For bipartite games, the asymptotic limit is taken not on changing Hilbert spaces directly but on a universal POVM $k$6-algebra $k$7; Banach–Alaoglu compactness yields weak-$k$8 convergent subsequences of positive functionals, and the Radon–Nikodym theorem for $k$9-algebras then reconstructs commuting POVM operators in the GNS representation (Kulpe et al., 2024). The multipartite extension replaces post-measurement states by chains of completely positive maps and instruments. There the main structural theorem states that correlations generated by sequential operationally no-signalling strategies are exactly commuting-operator correlations, and the proof introduces universal $Q$0-algebras of sequential PVMs together with a new chain rule for Radon–Nikodym derivatives of completely positive maps on $Q$1-algebras (Baroni et al., 16 Jul 2025).

On the optimization side, recent work formalizes sequential analogues of the NPA hierarchy. For bipartite compiled Bell games, the sequential NPA hierarchy is defined by an SDP over subnormalized moment matrices $Q$2 that track Bob-side polynomials up to degree $Q$3 while enforcing a strong no-signaling condition on the total Bob-side state (Klep et al., 22 Jul 2025). This hierarchy is asymptotically equivalent to standard NPA, is a relaxation of standard NPA at finite levels, admits a flatness criterion

$Q$4

and this flatness is equivalent to the existence of a finite-dimensional optimal quantum strategy for the underlying bipartite Bell game (Klep et al., 22 Jul 2025). The same paper identifies the conic dual as a sparse sum-of-squares hierarchy (Klep et al., 22 Jul 2025).

A complementary SDP/SoS approach restricts attention to “nice” certificates: sum-of-squares decompositions in which each square term only involves Alice operators for a single fixed question. The resulting one-sided NPA hierarchy searches exactly over such certificates and still converges to the commuting-operator value,

$Q$5

while any degree-1 SoS certificate can be transformed into a nice one (Cui et al., 23 Jul 2025). This framework recovers the earlier special-case bounds and also yields Kulpe et al.’s general asymptotic bound as a corollary (Cui et al., 23 Jul 2025).

6. Applications, adjacent constructions, and outstanding issues

Compiled nonlocal games have become a mechanism for classical verification of quantum behavior. Using compiled CHSH rigidity, Natarajan and Zhang constructed a single-prover cryptographically sound classical verification protocol for BQP. The protocol mirrors the functionality of Mahadev’s protocol, but its soundness proof follows the nonlocal analysis more directly and uses QFHE as a black box rather than explicitly invoking a trapdoor claw-free family or adaptive hardcore-bit assumption (Natarajan et al., 2023).

A more elaborate application appears in succinct arguments for QMA. Starting from a question-succinct self-test for Pauli measurements on maximally entangled states, the protocol compiles that nonlocal game into a single-prover classical-verifier argument system and proves soundness for the compiled mixed-basis Pauli test. The resulting protocol achieves completeness at least $Q$6, soundness at most a constant $Q$7, and communication $Q$8 (Metger et al., 2024). Unlike the earlier succinct QMA construction of Bartusek et al., which relied on post-quantum indistinguishability obfuscation, this protocol uses collapsing hash functions and a mild version of QHE, both stated to be implied by standard assumptions such as LWE (Metger et al., 2024).

The broader literature also contains related but distinct compilation motifs. One line compiles entanglement-network graphs into multipartite cooperating games: in the graphic-game framework, a network of independent EPR states is represented as a graph, players receive subgraphs as questions, and the payoff checks consistency conditions on shared vertices; the main criterion states that a graphic game has a quantum advantage if and only if $Q$9 (Luo, 2019). Another line introduces post-selection games, which add a discard option to ordinary nonlocal games and thereby compile possibilistic proofs such as Hardy’s paradox into conditionally scored games with computable local and Tsirelson bounds (Rodríguez et al., 26 Jan 2026).

Several limitations remain explicit in the literature. For games without finite-dimensional optimal strategies, the sequential-NPA approximation error may be unavoidable, and a universal bound of the form “compiled score $(q_1,\dots,q_k)\in (\{0,1\}^n)^k$0 true game value $(q_1,\dots,q_k)\in (\{0,1\}^n)^k$1 game-independent negligible term” is presented as potentially impossible in general (Klep et al., 22 Jul 2025). The same work points to correctness with auxiliary input for weakly commuting registers as a possible stronger cryptographic notion needed beyond the finite-dimensional regime (Klep et al., 22 Jul 2025). At the computational level, the moment-matrix size in the sequential-NPA analysis can grow like $(q_1,\dots,q_k)\in (\{0,1\}^n)^k$2, so only small hierarchy levels may be tractable unless the game has additional symmetry or sparsity (Klep et al., 22 Jul 2025). These caveats delimit the current frontier: the qualitative theory is now broad, including general bipartite and multipartite soundness, but the quantitative and computationally usable theory still depends strongly on game structure.