Separating ClonableQMA and QCMA Relative to a Classical Oracle

2026-10-05

AI Summary

This study addresses the open problem posed by Nehoran and Zhandry regarding whether clonable quantum proofs retain a computational advantage over classical proofs. By integrating computational complexity theory, oracle construction techniques, and quantum information theory, this work constructs a specific classical oracle O to investigate the separation between the complexity classes QCMA and ClonableQMA. It achieves the first rigorous separation between clonable quantum proofs and classical proofs in the classical oracle setting. Specifically, this paper establishes that QCMA^O ≠ ClonableQMA^O and demonstrates a classical oracle separation between BQP/clonableqpoly and BQP/poly. These results confirm the computational superiority of clonable quantum proofs and reveal their potential applications in quantum cryptography.

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.

Problem

• QMA
• QCMA
• ClonableQMA
• classical oracle
• unclonability

Innovation

• ClonableQMA
• QCMA
• Classical Oracle Separation
• Quantum Proofs
• Unclonability