 
Summary: Quantum Versus Classical Proofs and Advice
Scott Aaronson # Greg Kuperberg +
Abstract
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 QCMA. More concretely, we show that any quantum algorithm
needs# ``q 2 n
m+1 '' queries
to find an nqubit ``marked state'' ##, even if given an mbit classical description of ## together with a quantum
black box that recognizes ##. Second, we give an explicit QCMA protocol that nearly achieves this lower bound.
Third, we show that, in the one previouslyknown case where quantum proofs seemed to provide an exponential
advantage, classical proofs are basically just as powerful. In particular, Watrous gave a QMA protocol for verifying
nonmembership in finite groups. Under plausible grouptheoretic assumptions, we give a QCMA protocol for the
same problem. Even with no assumptions, our protocol makes only polynomially 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.
1 Introduction
If someone hands you a quantum state, is that more ``useful'' than being handed a classical string with a comparable
number of bits? In particular, are there truths that you can efficiently verify, and are there problems that you can
efficiently solve, using the quantum state but not using the string? These are the questions that this paper addresses,
