arXiv:2610.06740

Separating ClonableQMA and QCMA Relative to a Classical Oracle

Alper Cakan, Kai-Min Chung, Wei-Hsiang Hung, Tzu-Yi Yang

cs.CCquant-phSubmitted Oct 5, 2026• Announced Oct 6, 2026• 0 citations

Abstract

Since the introduction of the complexity class QMA as a quantum-verifier analogue of NP (Kitaev, 1997), many have wondered whether quantum proofs are necessary or classical proofs suffice - that is, whether QMA = QCMA or QCMA != QMA (Aharonov and Naveh, 2002; Aaronson and Kuperberg, CCC '07). This longstanding question was recently answered by works of Bostanci, Haferkamp, Nirkhe, and Zhandry (STOC '26) and Bostanci, Huang, and Vaikuntanathan (FOCS '26), which showed that quantum proofs are more powerful than classical ones in the classical-oracle setting. However, it remains unclear what exactly makes quantum proofs more powerful than classical ones. In the information-theoretic setting, a family of quantum states is not classicalizable if and only if it is unclonable. Indeed, these recent works also explicitly highlight the unclonability of their quantum proofs as a key mechanism behind their separations, and their arguments crucially rely on this property. This raises the question of whether unclonability is necessary for quantum proofs to be more powerful than classical ones. In this work, we show that even clonable quantum proofs can be more powerful than classical ones relative to a classical oracle by constructing a classical oracle O such that QCMA^O != ClonableQMA^O. This resolves the open question of Nehoran and Zhandry (ITCS '24), who established the analogous separation relative to a quantum oracle. We also show a classical-oracle separation between BQP/clonableqpoly and BQP/poly, and give applications of our results to quantum cryptography.

Related papers

Separating QMA from QCIP with a Classical Oracle, or, the Power of Quantum Proofs over Classical Interaction for Quantum Verifiers

Alper Çakan

arXiv:2610.03648Oct 2, 20260 citations

Separating Quantum and Classical Advice with Good Codes

John Bostanci, Andrew Huang, Vinod Vaikuntanathan

quant-pharXiv:2602.09385Feb 11, 20260 citations

Classical vs Quantum Advice and Proofs Under Classically-Accessible Oracle

Xingjian Li, Liu, Qipeng, Angelos Pelecanos +1 more

arXiv:2303.04298Leibniz international proceedings in informaticsMar 8, 20232 citations

A Computational Separation Between Quantum No-Cloning and No-Telegraphing

Barak Nehoran, Mark Zhandry

arXiv:2302.01858Leibniz international proceedings in informaticsFeb 3, 20236 citations

QMA vs QCMA and Pseudorandomness

Jiahui Liu, Saachi Mutreja, Henry Yuen

arXiv:2411.14416ACM Symposium on Theory of Computing (STOC)Jun 15, 20252 citations

A Relativizing MIP for BQP

Scott Aaronson, Anand Natarajan, Avishay Tal +1 more

quant-pharXiv:2604.11952Apr 15, 20260 citations

arXiv:2610.06740

Separating ClonableQMA and QCMA Relative to a Classical Oracle

Alper Cakan, Kai-Min Chung, Wei-Hsiang Hung, Tzu-Yi Yang

cs.CCquant-phSubmitted Oct 5, 2026• Announced Oct 6, 2026• 0 citations

arXiv HTML PDF

Abstract

However, it remains unclear what exactly makes quantum proofs more powerful than classical ones. In the information-theoretic setting, a family of quantum states is not classicalizable if and only if it is unclonable. Indeed, these recent works also explicitly highlight the unclonability of their quantum proofs as a key mechanism behind their separations, and their arguments crucially rely on this property. This raises the question of whether unclonability is necessary for quantum proofs to be more powerful than classical ones.

In this work, we show that even clonable quantum proofs can be more powerful than classical ones relative to a classical oracle by constructing a classical oracle O such that QCMA^O != ClonableQMA^O. This resolves the open question of Nehoran and Zhandry (ITCS '24), who established the analogous separation relative to a quantum oracle. We also show a classical-oracle separation between BQP/clonableqpoly and BQP/poly, and give applications of our results to quantum cryptography.

Related papers

Separating QMA from QCIP with a Classical Oracle, or, the Power of Quantum Proofs over Classical Interaction for Quantum Verifiers

Alper Çakan

Whether some problems require quantum proofs has been a central question in quantum complexity (Aharonov and Naveh, 2002; Aaronson and Kuperberg, CCC 2007). Recently, breakthrough work of Bostanci, Haferkamp, Nirkhe, and Zhandry (STOC 2026), followed by a simpler separation due to Bostanci, Huang, and Vaikuntanathan (FOCS 2026), established a classical oracle separation between $\mathsf{QMA}$ and $\mathsf{QCMA}$ . However, while they are not in $\mathsf{QCMA}$ , the problems used in both separations still lie in $\mathsf{AM}$ : they admit a two-message public-coin proof system with a classical verifier. In this work, we ask whether some problems truly require quantum proofs. More formally, we consider the complexity class $\mathsf{QCIP}$ , introduced by Buhrman, Le Gall, and Weggemans (2024), where an efficient quantum verifier interacts with an unbounded prover over a classical channel for an arbitrary polynomial number of rounds. We construct a classical oracle $\mathcal{O}$ such that $\mathsf{QMA}^{\mathcal{O}}\not\subseteq\mathsf{QCIP}^{\mathcal{O}}$ , thus showing that some languages indeed require quantum proofs, with no classical replacements. Since $\mathsf{QCMA}=\mathsf{QCIP}[1]$ , this strengthens the earlier $\mathsf{QMA}$ -- $\mathsf{QCMA}$ separations, which now follow as a special case of our result. As a technical contribution, we extend to the complexity theory setting the techniques developed by Cakan, Goyal, and Shmueli (CRYPTO 2026) in the context of cryptography for analyzing classically interacting quantum machines. We believe this may be of independent interest.Whether some problems require quantum proofs has been a central question in quantum complexity (Aharonov and Naveh, 2002; Aaronson and Kuperberg, CCC 2007). Recently, breakthrough work of Bostanci, Haferkamp, Nirkhe, and Zhandry (STOC 2026), followed by a simpler separation due to Bostanci, Huang, and Vaikuntanathan (FOCS 2026), established a classical oracle separation between $\mathsf{QMA}$ and $\mathsf{QCMA}$. However, while they are not in $\mathsf{QCMA}$, the problems used in both separations still lie in $\mathsf{AM}$: they admit a two-message public-coin proof system with a classical verifier. In this work, we ask whether some problems truly require quantum proofs. More formally, we consider the complexity class $\mathsf{QCIP}$, introduced by Buhrman, Le Gall, and Weggemans (2024), where an efficient quantum verifier interacts with an unbounded prover over a classical channel for an arbitrary polynomial number of rounds. We construct a classical oracle $\mathcal{O}$ such that $\mathsf{QMA}^{\mathcal{O}}\not\subseteq\mathsf{QCIP}^{\mathcal{O}}$, thus showing that some languages indeed require quantum proofs, with no classical replacements. Since $\mathsf{QCMA}=\mathsf{QCIP}[1]$, this strengthens the earlier $\mathsf{QMA}$--$\mathsf{QCMA}$ separations, which now follow as a special case of our result. As a technical contribution, we extend to the complexity theory setting the techniques developed by Cakan, Goyal, and Shmueli (CRYPTO 2026) in the context of cryptography for analyzing classically interacting quantum machines. We believe this may be of independent interest.

arXiv:2610.03648Oct 2, 20260 citations

Separating Quantum and Classical Advice with Good Codes

John Bostanci, Andrew Huang, Vinod Vaikuntanathan

We show an unconditional classical oracle separation between the class of languages that can be verified using a quantum proof ( $\mathsf{QMA}$ ) and the class of languages that can be verified with a classical proof ( $\mathsf{QCMA}$ ). Compared to the recent work of Bostanci, Haferkamp, Nirkhe, and Zhandry (STOC 2026), our proof is conceptually and technically simpler, and readily extends to other oracle separations. In particular, our techniques yield the first unconditional classical oracle separation between the class of languages that can be decided with quantum advice ( $\mathsf{BQP}/\mathsf{qpoly}$ ) and the class of languages that can be decided with classical advice ( $\mathsf{BQP}/\mathsf{poly}$ ), improving on the quantum oracle separation of Aaronson and Kuperberg (CCC 2007) and the classically-accessible classical oracle separation of Li, Liu, Pelecanos and Yamakawa (ITCS 2024). Our oracles are based on the code intersection problem introduced by Yamakawa and Zhandry (FOCS 2022), combined with codes that have extremely good list-recovery properties.We show an unconditional classical oracle separation between the class of languages that can be verified using a quantum proof ($\mathsf{QMA}$) and the class of languages that can be verified with a classical proof ($\mathsf{QCMA}$). Compared to the recent work of Bostanci, Haferkamp, Nirkhe, and Zhandry (STOC 2026), our proof is conceptually and technically simpler, and readily extends to other oracle separations. In particular, our techniques yield the first unconditional classical oracle separation between the class of languages that can be decided with quantum advice ($\mathsf{BQP}/\mathsf{qpoly}$) and the class of languages that can be decided with classical advice ($\mathsf{BQP}/\mathsf{poly}$), improving on the quantum oracle separation of Aaronson and Kuperberg (CCC 2007) and the classically-accessible classical oracle separation of Li, Liu, Pelecanos and Yamakawa (ITCS 2024). Our oracles are based on the code intersection problem introduced by Yamakawa and Zhandry (FOCS 2022), combined with codes that have extremely good list-recovery properties.

quant-pharXiv:2602.09385Feb 11, 20260 citations

Classical vs Quantum Advice and Proofs Under Classically-Accessible Oracle

Xingjian Li, Liu, Qipeng, Angelos Pelecanos +1 more

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.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.

arXiv:2303.04298Leibniz international proceedings in informaticsMar 8, 20232 citations

A Computational Separation Between Quantum No-Cloning and No-Telegraphing

Barak Nehoran, Mark Zhandry

Two of the fundamental no-go theorems of quantum information are the no-cloning theorem (that it is impossible to make copies of general quantum states) and the no-teleportation theorem (the prohibition on telegraphing, or sending quantum states over classical channels without pre-shared entanglement). They are known to be equivalent, in the sense that a collection of quantum states is telegraphable if and only if it is clonable. Our main result suggests that this is not the case when computational efficiency is considered. We give a collection of quantum states and quantum oracles relative to which these states are efficiently clonable but not efficiently telegraphable. Given that the opposite scenario is impossible (states that can be telegraphed can always trivially be cloned), this gives the most complete quantum oracle separation possible between these two important no-go properties. We additionally study the complexity class clonableQMA, a subset of QMA whose witnesses are efficiently clonable. As a consequence of our main result, we give a quantum oracle separation between clonableQMA and the class QCMA, whose witnesses are restricted to classical strings. We also propose a candidate oracle-free promise problem separating these classes. We finally demonstrate an application of clonable-but-not-telegraphable states to cryptography, by showing how such states can be used to protect against key exfiltration.Two of the fundamental no-go theorems of quantum information are the no-cloning theorem (that it is impossible to make copies of general quantum states) and the no-teleportation theorem (the prohibition on telegraphing, or sending quantum states over classical channels without pre-shared entanglement). They are known to be equivalent, in the sense that a collection of quantum states is telegraphable if and only if it is clonable. Our main result suggests that this is not the case when computational efficiency is considered. We give a collection of quantum states and quantum oracles relative to which these states are efficiently clonable but not efficiently telegraphable. Given that the opposite scenario is impossible (states that can be telegraphed can always trivially be cloned), this gives the most complete quantum oracle separation possible between these two important no-go properties. We additionally study the complexity class clonableQMA, a subset of QMA whose witnesses are efficiently clonable. As a consequence of our main result, we give a quantum oracle separation between clonableQMA and the class QCMA, whose witnesses are restricted to classical strings. We also propose a candidate oracle-free promise problem separating these classes. We finally demonstrate an application of clonable-but-not-telegraphable states to cryptography, by showing how such states can be used to protect against key exfiltration.

arXiv:2302.01858Leibniz international proceedings in informaticsFeb 3, 20236 citations

QMA vs QCMA and Pseudorandomness

Jiahui Liu, Saachi Mutreja, Henry Yuen

We study a longstanding question of Aaronson and Kuperberg on whether there exists a classical oracle separating $\mathsf{QMA}$ from $\mathsf{QCMA}$ . Settling this question in either direction would yield insight into the power of quantum proofs over classical proofs. We show that such an oracle exists if a certain quantum pseudorandomness conjecture holds. Roughly speaking, the conjecture posits that quantum algorithms cannot, by making few queries, distinguish between the uniform distribution over permutations versus permutations drawn from so-called "dense" distributions. Our result can be viewed as establishing a "win-win" scenario: either there is a classical oracle separation of $\mathsf{QMA}$ from $\mathsf{QCMA}$ , or there is quantum advantage in distinguishing pseudorandom distributions on permutations.We study a longstanding question of Aaronson and Kuperberg on whether there exists a classical oracle separating $\mathsf{QMA}$ from $\mathsf{QCMA}$. Settling this question in either direction would yield insight into the power of quantum proofs over classical proofs. We show that such an oracle exists if a certain quantum pseudorandomness conjecture holds. Roughly speaking, the conjecture posits that quantum algorithms cannot, by making few queries, distinguish between the uniform distribution over permutations versus permutations drawn from so-called "dense" distributions. Our result can be viewed as establishing a "win-win" scenario: either there is a classical oracle separation of $\mathsf{QMA}$ from $\mathsf{QCMA}$, or there is quantum advantage in distinguishing pseudorandom distributions on permutations.

arXiv:2411.14416ACM Symposium on Theory of Computing (STOC)Jun 15, 20252 citations

A Relativizing MIP for BQP

Scott Aaronson, Anand Natarajan, Avishay Tal +1 more

Complexity class containments involving interactive proof classes are famously nonrelativizing: although $\mathsf{IP} = \mathsf{PSPACE}$ , Fortnow and Sipser showed that that there exists an oracle relative to which $\mathsf{coNP} \not\subseteq \mathsf{IP}$ . In contrast, the question of whether the containment $\mathsf{BQP} \subseteq \mathsf{IP}$ is relativizing remains wide open. In this work we make progress towards resolving this question by showing that the containment $\mathsf{BQP} \subseteq \mathsf{MIP}$ holds with respect to any classical oracle. We obtain this result by constructing, for any classical oracle $O$ , a $\mathsf{PCP}$ proof system for $\mathsf{BQP}^{O}$ where the verifier makes polynomially many classical queries to an exponentially-long proof, and to the oracle $O$ . Our construction is inspired by the state synthesis algorithm of Grover and Rudolph, and serves as a complement to the "exponential PCP" constructed by Aharonov, Arad, and Vidick, which achieves similar parameters but which is based on different ideas and does not relativize. We propose relativization as a proxy for prover efficiency, and hope that progress towards an $\mathsf{IP}$ for $\mathsf{BQP}$ in the oracle world will lead to a non-cryptographic interactive protocol for proving any quantum computation to a classical skeptic in the unrelativized world, which is a longstanding open problem in quantum complexity theory.Complexity class containments involving interactive proof classes are famously nonrelativizing: although $\mathsf{IP} = \mathsf{PSPACE}$, Fortnow and Sipser showed that that there exists an oracle relative to which $\mathsf{coNP} \not\subseteq \mathsf{IP}$. In contrast, the question of whether the containment $\mathsf{BQP} \subseteq \mathsf{IP}$ is relativizing remains wide open. In this work we make progress towards resolving this question by showing that the containment $\mathsf{BQP} \subseteq \mathsf{MIP}$ holds with respect to any classical oracle. We obtain this result by constructing, for any classical oracle $O$, a $\mathsf{PCP}$ proof system for $\mathsf{BQP}^{O}$ where the verifier makes polynomially many classical queries to an exponentially-long proof, and to the oracle $O$. Our construction is inspired by the state synthesis algorithm of Grover and Rudolph, and serves as a complement to the "exponential PCP" constructed by Aharonov, Arad, and Vidick, which achieves similar parameters but which is based on different ideas and does not relativize. We propose relativization as a proxy for prover efficiency, and hope that progress towards an $\mathsf{IP}$ for $\mathsf{BQP}$ in the oracle world will lead to a non-cryptographic interactive protocol for proving any quantum computation to a classical skeptic in the unrelativized world, which is a longstanding open problem in quantum complexity theory.

quant-pharXiv:2604.11952Apr 15, 20260 citations

\mathsf{QMA}

and

\mathsf{QCMA}

. However, while they are not in

\mathsf{QCMA}

, the problems used in both separations still lie in

\mathsf{AM}

: they admit a two-message public-coin proof system with a classical verifier. In this work, we ask whether some problems truly require quantum proofs. More formally, we consider the complexity class

\mathsf{QCIP}

, introduced by Buhrman, Le Gall, and Weggemans (2024), where an efficient quantum verifier interacts with an unbounded prover over a classical channel for an arbitrary polynomial number of rounds. We construct a classical oracle

\mathcal{O}

such that

\mathsf{QMA}^{\mathcal{O}}\not\subseteq\mathsf{QCIP}^{\mathcal{O}}

, thus showing that some languages indeed require quantum proofs, with no classical replacements. Since

\mathsf{QCMA}=\mathsf{QCIP}[1]

, this strengthens the earlier

\mathsf{QMA}

\mathsf{QCMA}

separations, which now follow as a special case of our result. As a technical contribution, we extend to the complexity theory setting the techniques developed by Cakan, Goyal, and Shmueli (CRYPTO 2026) in the context of cryptography for analyzing classically interacting quantum machines. We believe this may be of independent interest.

We show an unconditional classical oracle separation between the class of languages that can be verified using a quantum proof (

\mathsf{QMA}

) and the class of languages that can be verified with a classical proof (

\mathsf{QCMA}

). Compared to the recent work of Bostanci, Haferkamp, Nirkhe, and Zhandry (STOC 2026), our proof is conceptually and technically simpler, and readily extends to other oracle separations. In particular, our techniques yield the first unconditional classical oracle separation between the class of languages that can be decided with quantum advice (

\mathsf{BQP}/\mathsf{qpoly}

) and the class of languages that can be decided with classical advice (

\mathsf{BQP}/\mathsf{poly}

), improving on the quantum oracle separation of Aaronson and Kuperberg (CCC 2007) and the classically-accessible classical oracle separation of Li, Liu, Pelecanos and Yamakawa (ITCS 2024). Our oracles are based on the code intersection problem introduced by Yamakawa and Zhandry (FOCS 2022), combined with codes that have extremely good list-recovery properties.

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.

We study a longstanding question of Aaronson and Kuperberg on whether there exists a classical oracle separating

\mathsf{QMA}

from

\mathsf{QCMA}

. Settling this question in either direction would yield insight into the power of quantum proofs over classical proofs. We show that such an oracle exists if a certain quantum pseudorandomness conjecture holds. Roughly speaking, the conjecture posits that quantum algorithms cannot, by making few queries, distinguish between the uniform distribution over permutations versus permutations drawn from so-called "dense" distributions. Our result can be viewed as establishing a "win-win" scenario: either there is a classical oracle separation of

\mathsf{QMA}

from

\mathsf{QCMA}

, or there is quantum advantage in distinguishing pseudorandom distributions on permutations.

Complexity class containments involving interactive proof classes are famously nonrelativizing: although

\mathsf{IP} = \mathsf{PSPACE}

, Fortnow and Sipser showed that that there exists an oracle relative to which

\mathsf{coNP} \not\subseteq \mathsf{IP}

. In contrast, the question of whether the containment

\mathsf{BQP} \subseteq \mathsf{IP}

is relativizing remains wide open. In this work we make progress towards resolving this question by showing that the containment

\mathsf{BQP} \subseteq \mathsf{MIP}

holds with respect to any classical oracle. We obtain this result by constructing, for any classical oracle

O

, a

\mathsf{PCP}

proof system for

\mathsf{BQP}^{O}

where the verifier makes polynomially many classical queries to an exponentially-long proof, and to the oracle

O

. Our construction is inspired by the state synthesis algorithm of Grover and Rudolph, and serves as a complement to the "exponential PCP" constructed by Aharonov, Arad, and Vidick, which achieves similar parameters but which is based on different ideas and does not relativize. We propose relativization as a proxy for prover efficiency, and hope that progress towards an

\mathsf{IP}

for

\mathsf{BQP}

in the oracle world will lead to a non-cryptographic interactive protocol for proving any quantum computation to a classical skeptic in the unrelativized world, which is a longstanding open problem in quantum complexity theory.