En Route to a Standard QMA1 vs. QCMA Oracle Separation

Abstract: We study the power of quantum witnesses under perfect completeness. We construct a classical oracle relative to which a language lies in $\mathsf{QMA}_1$ but not in $\mathsf{QCMA}$ when the $\mathsf{QCMA}$ verifier is only allowed polynomially many adaptive rounds and exponentially many parallel queries per round. Additionally, we derandomize the permutation-oracle separation of Fefferman and Kimmel, obtaining an in-place oracle separation between $\mathsf{QMA}_1$ and $\mathsf{QCMA}$. Furthermore, we focus on $\mathsf{QCMA}$ and $\mathsf{QMA}$ with an exponentially small gap, where we show a separation assuming the gap is fixed, but not when it may be arbitrarily small. Finally, we derive consequences for approximate ground-state preparation from sparse Hamiltonian oracle access, including a bounded-adaptivity frustration-free variant.

• The paper introduces a bounded-adaptivity oracle separation for QMA₁ using pointer-chasing permutations that resist any bounded-round QCMA protocol.
• It derives a derandomized in-place oracle separation by replacing random permutations with explicit deterministic constructions, ensuring perfect completeness.
• The work demonstrates that quantum proofs enforce superpolynomial circuit lower bounds for sparse Hamiltonian ground state preparation, linking proof power to state complexity.

QMA₁ vs. QCMA Oracle Separations: Techniques, Results, and Implications

Introduction and Context

The question of whether quantum witnesses provide additional power over classical witnesses in quantum proof systems is a central problem in quantum complexity theory. The classes QMA (Quantum Merlin-Arthur) and its classical-witness analogue QCMA are widely studied, with the inclusion $QCMA \subseteq QMA$ being obvious by definition. The core challenge is to determine if this inclusion is strict, especially in "standard" settings—i.e., where the verification is performed by a polynomial-time quantum algorithm with black-box access to a classical or unitary oracle.

While numerous oracle separations have illustrated quantum advantage in various settings, most fail to address perfect completeness (QMA₁) or do not operate in natural classical oracle models. Notably, previous separations using quantum oracles [aaronson2007quantum] and randomized in-place permutation oracles [FK18] left open whether QMA₁ could be separated from QCMA in the presence of a "standard" (classical) oracle and/or under perfect completeness constraints. Recent works have closed the gap for QMA vs. QCMA (Bostanci et al., 12 Nov 2025, Bostanci et al., 10 Feb 2026), yet perfect completeness and circuit lower bounds still presented significant technical challenges.

This paper introduces new techniques to address these challenges and provides several separations between QMA₁ and QCMA variants in increasingly standard and constrained oracle models, explores vanishing completeness-soundness gaps, and draws structural consequences for sparse Hamiltonian ground state complexity.

Main Results and Technical Innovations

Bounded Adaptivity Separations

Bounded-Adaptivity Oracle Separation:

The paper constructs, for any fixed polynomial $R$, a language relative to a classical oracle where $L \in QMA_1^O$ but $L \notin QCMA^O$, provided the QCMA verifier is limited to $R$ adaptive rounds (with potentially exponential parallel queries in each round). Crucially, this separation persists even as the number of queries per round is allowed to grow exponentially, matching settings considered in prior works [BK24]. The result leverages a pointer-chasing permutation problem designed to have an efficiently checkable witness for $QMA_1$ (based on the uniform superposition over a certain preimage set) but that resists any QCMA protocol of bounded adaptive query depth.

The pointer-chasing construction ensures that—even with exponentially many queries per round—the classical witness cannot efficiently encode the advice needed for parity detection over hidden structure when only a bounded number of adaptive updates to the transcript are permitted. The associated lower bounds leverage hybrid techniques and heavy-path inductive arguments quantifying indistinguishability of oracles with matching query transcripts.

In-Place Oracle and Derandomization

Derandomized In-Place Oracle Separation:

The authors derandomize the Fefferman-Kimmel randomized permutation oracle separation [FK18], obtaining an in-place oracle (i.e., reversible permutation oracle, no access to the inverse) relative to which $QMA_1$ is separated from QCMA—again even against QCMA verifiers with $2^{o(n)}$ rounds and exponential queries per round. This advances the state-of-the-art by replacing random permutations with explicit, deterministic pointer-chasing instances, and strengthens prior randomized and non-perfect-completeness results.

The key technical device is a transcript analysis that records, at each depth, the information "learnable" by any QCMA protocol—but which, by design, is insufficient for distinguishing YES from NO instances. This is formalized via construction of canonical completions and the heavy-point method, bounding the probability that sufficient distinguishing information is revealed in any transcript sequence.

Vanishing Gap and Precise Verification

Separation at Exponentially Small Gaps:

For verification protocols with exponentially small completeness-soundness gaps, the paper establishes that, when the gap is fixed at $2^{-n}$, a classical oracle separation between QMA and QCMA still holds. However, they show that when one considers the union over all (arbitrarily small) gaps—i.e., PreciseQCMA—this separation becomes insurmountable: their problem lies in both PreciseQCMA and the quantum advice class, but not in $QCMA(2^{-n})$. This suggests the absence of efficient gap amplification procedures for the precise variants, as otherwise gap amplification would collapse the aforementioned separation.

The adaptation modifies the combinatorial direct product and code-intersection hardness machinery of (Bostanci et al., 10 Feb 2026), carefully tuning soundness parameters to work in the vanishing gap regime.

Ground State Complexity from Oracle Separations

Circuit Complexity of Ground States:

By applying Kitaev's circuit-to-Hamiltonian construction to their QMA₁ verifiers and the resulting hard pointer-chasing problems, the authors lift their oracle separation to derive lower bounds on the circuit complexity of preparing approximate ground states of families of $R$0-sparse Hamiltonians specified via black-box (oracle) access. Specifically, any quantum circuit with access to the oracle that prepares a low-energy state must have superpolynomial size. This forms an oracle analogue to the NLTS theorem [ABN23], relating proof-theoretic complexity to ground-state complexity for sparse Hamiltonians.

A related extension shows a frustration-free variant (where the ground state can be prepared efficiently by an adaptive quantum circuit, but only with a number of adaptive rounds exceeding those permitted to the QCMA adversary).

Numerical Bounds, Claims, and Contrasts

• The pointer-chasing in-place separation withstands QCMA adversaries with up to $R$1 parallel queries and $R$2 adaptivity rounds for any $R$3, with the hardness preserved as soon as query transcript size remains $R$4.
• The explicit probability bounds for bad transcripts are double-exponentially small in $R$5, ensuring that diagonalization over all QCMA verifiers is effective.
• For precise (exponentially small gap) separation, achievable success probability for the no-query classical algorithm is no more than $R$6, while the lower bound for a (non-existent) QCMA protocol exceeds $R$7 for polynomial $R$8.
• Ground state circuits smaller than $R$9 cannot achieve energy within $L \in QMA_1^O$0 of the ground state, relative to the oracle, and frustration-free versions hold under matching query-depth constraints.

Implications and Future Directions

On the practical side, these results clarify the structural strength imparted by quantum proofs even in settings where completeness is perfect, complexity parameters are adversarial, and QCMA protocols are allowed aggressive query and parallelization strategies. They also solidify the pointer-chasing paradigm as yielding the sharpest current methods for analyzing oracle (and possibly unrelativized) QMA₁ vs. QCMA separations.

The connection to the complexity of ground-state preparation reinforces the intuition that quantum proofs can encode quantum advice that is fundamentally "preparation-hard" for any QCMA protocol—especially in sparse Hamiltonian simulation architectures. These oracle-based bounds directly motivate the search for unrelativized, explicit families with analogous ground state complexity properties, with implications for both quantum PCP and quantum supremacy questions.

On the theoretical front, the limitations exposed with regard to gap amplification in PreciseQMA/PreciseQCMA bolster the viewpoint that the precise setting forms an ultimate quantum analogue of PSPACE-hardness, and further study in this regime is warranted. The explicit transcript and diagonalization techniques developed here may be applicable for analyzing other settings where adaptivity, query parallelism, and proof type intersect.

Finally, as the construction techniques are independent of recent parallel work (Bostanci et al., 12 Nov 2025, Bostanci et al., 10 Feb 2026), the combination of ideas here points to new strategies for constructing fine-grained separations—especially in conjunction with cryptographic and coding-theoretic primitives underlying modern oracle lower bounds.

Conclusion

This work completes several open directions regarding standard classical and in-place oracle separations between QMA₁ and QCMA. It introduces bounded-adaptivity pointer-chasing frameworks, derandomized explicit models, extensions to precise-gapped settings, and transposes the resulting complexity separations to nontrivial quantum circuit lower bounds for sparse Hamiltonian ground states. The results push forward our understanding of the relative power of quantum and classical witnesses, and provide a toolkit likely to inform future developments at the intersection of quantum complexity, circuit lower bounds, and quantum PCP-type conjectures.

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References (Represented by arXiv IDs for key prior/parallel results)

• [aaronson2007quantum]: "Quantum versus classical proofs and advice"
• [FK18]: "Quantum vs. Classical Proofs and Subset Verification"
• (Bostanci et al., 12 Nov 2025): "Separating QMA from QCMA with a classical oracle"
• (Bostanci et al., 10 Feb 2026): "Separating Quantum and Classical Advice with Good Codes"
• [BK24]: "Oracle Separation of QMA and QCMA with Bounded Adaptivity"
• [ABN23]: "NLTS Hamiltonians from Good Quantum Codes"