Quantum versus Classical Proofs and Advice

Quantum computing sceptics have claimed that any theory involving
exponentially-long vectors is inherently unreasonable.  But if someone hands
you a quantum state, is that really like being handed an exponential amount
of classical information?  I'll describe recent results suggesting that the
answer, at least for complexity-theory purposes, is "no."  These results
include simulations of quantum advice using classical advice (e.g. BQP/qpoly
is contained in PP/poly, and QMA/qpoly is contained in PSPACE/poly), as well
as a conditional "dequantization" of John Watrous's quantum proof protocol
for the Group Non-Membership problem.