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arXiv:2609.02865

A quantum oracle separation between QMA(2) and QMA

John Bostanci, Sabee Grewal, Jonas Haferkamp, Andrew Huang, Yeongwoo Hwang, Anand Natarajan, Chinmay Nirkhe

quant-phSubmitted Sep 2, 2026• Announced Sep 3, 2026• 0 citations
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Abstract

We find a quantum oracle relative to which $\mathsf{QMA} \neq \mathsf{QMA}(2)$. As a consequence, we resolve the no-disentanglers conjecture of Watrous: for every $ε+δ<1$, any $(ε,δ)$-disentangler requires input size exponential in the number of output qubits. Our proof combines the unitarily invariant polynomial method of She and Yuen (ITCS '23) with a new construction based on the symmetric and antisymmetric subspace projectors, reducing the $\mathsf{QMA}$ lower bound to the approximate degree of $\mathrm{OR}$.

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We find a quantum oracle relative to which QMA≠QMA(2)\mathsf{QMA} \neq \mathsf{QMA}(2)QMA=QMA(2). As a consequence, we resolve the no-disentanglers conjecture of Watrous: for every ε+δ<1ε+δ<1ε+δ<1, any (ε,δ)(ε,δ)(ε,δ)-disentangler requires input size exponential in the number of output qubits. Our proof combines the unitarily invariant polynomial method of She and Yuen (ITCS '23) with a new construction based on the symmetric and antisymmetric subspace projectors, reducing the QMA\mathsf{QMA}QMA lower bound to the approximate degree of OR\mathrm{OR}OR.
We exhibit a classical oracle relative to which QMA \neq QMA(2). The separation is based on a separable Hamiltonian problem defined by signed edge constraints on a graph. The YES instances have zero-energy product states, while the NO instances, constructed using the antisymmetric subspace, have entangled zero-energy states but no low-energy product states. For the QMA lower bound, we use compressed-oracle techniques to track the information learned by a quantum computation making few queries to a YES instance. The key step is to map the resulting superpositions of partial oracle records to the NO family without significantly changing the verifier's behavior.
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We give an oracle separation between QMA and QCMA for quantum algorithms that have bounded adaptivity in their oracle queries; that is, the number of rounds of oracle calls is small, though each round may involve polynomially many queries in parallel. Our oracle construction is a simplified version of the construction used recently by Li, Liu, Pelecanos, and Yamakawa (2023), who showed an oracle separation between QMA and QCMA when the quantum algorithms are only allowed to access the oracle classically. To prove our results, we introduce a property of relations called \emph{slipperiness}, which may be useful for getting a fully general classical oracle separation between QMA and QCMA.
We find a modification to QMA where having one quantum proof is strictly less powerful than having two unentangled proofs, assuming EXP ≠\ne= NEXP. This gives a new route to prove QMA(2) = NEXP that overcomes the primary drawback of a recent approach [arXiv:2402.18790 , arXiv:2306.13247] (QIP 2024). Our modification endows each proof with a form of *multipartite* unentanglement: after tracing out one register, a small number of qubits are separable from the rest of the state.
It is a long-standing open question to construct a classical oracle relative to which BQP/qpoly ≠\neq= BQP/poly or QMA ≠\neq= QCMA. In this paper, we construct classically-accessible classical oracles relative to which BQP/qpoly ≠\neq= BQP/poly and QMA ≠\neq= QCMA. Here, classically-accessible classical oracles are oracles that can be accessed only classically even for quantum algorithms. Based on a similar technique, we also show an alternative proof for the separation of QMA and QCMA relative to a distributional quantumly-accessible classical oracle, which was recently shown by Natarajan and Nirkhe.
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