Quantum versus Classical Proofs and Advice

This paper studies whether quantum proofs are more powerful than classical proofs, or in complexity terms, whether QMA = QCMA. We prove three results about this question. First, we give a "quantum oracle separation" between QMA and QCM A. More concretely, we show that any quantum algorithm needs Omega (radic²ⁿ-_m+1) queries to find an n-qubit "marked state" Psi rang, even if given an m-bit classical description of Psi rang together with a quantum black box that recognizes Psi rang. Second, we give an explicit QCMA protocol that nearly achieves this lower bound. Third, we show that, in the one previously-known case where quantum proofs seemed to provide an exponential advantage, classical proofs are basically just as powerful. In particular, Wa- trous gave a QM IK protocol for verifying non-membership infinite groups. Under plausible group-theoretic assumptions, we give a QCMA protocol for the same problem. Even with no assumptions, our protocol makes only poly-nomially many queries to the group oracle. We end with some conjectures about quantum versus classical oracles, and about the possibility of a classical oracle separation between QMA and QCMA.