  Let L be a language decided by a constant-round quantum Arthur-Merlin (QAM)
protocol with negligible soundness error and all but possibly the last message
being classical. We prove that if this protocol is zero knowledge with a
black-box, quantum simulator S, then L in BQP. Our result also applies to any
language having a three-round quantum interactive proof (QIP), with all but
possibly the last message being classical, with negligible soundness error and
a black-box quantum simulator.
  These results in particular make it unlikely that certain protocols can be
composed in parallel in order to reduce soundness error, while maintaining zero
knowledge with a black-box quantum simulator. They generalize analogous
classical results of Goldreich and Krawczyk (1990).
  Our proof goes via a reduction to quantum black-box search. We show that the
existence of a black-box quantum simulator for such protocols when L notin BQP
would imply an impossibly-good quantum search algorithm.
