  We present three contributions to the understanding of QMA with multiple
provers:
  1) We give a tight soundness analysis of the protocol of [Blier and Tapp,
ICQNM '09], yielding a soundness gap Omega(1/N^2). Our improvement is achieved
without the use of an instance with a constant soundness gap (i.e., without
using a PCP).
  2) We give a tight soundness analysis of the protocol of [Chen and Drucker,
ArXiV '10], thereby improving their result from a monolithic protocol where
Theta(sqrt(N)) provers are needed in order to have any soundness gap, to a
protocol with a smooth trade-off between the number of provers k and a
soundness gap Omega(k^2/N), as long as k>=Omega(log N). (And, when
k=Theta(sqrt(N)), we recover the original parameters of Chen and Drucker.)
  3) We make progress towards an open question of [Aaronson et al., ToC '09]
about what kinds of NP-complete problems are amenable to sublinear
multiple-prover QMA protocols, by observing that a large class of such examples
can easily be derived from results already in the PCP literature - namely, at
least the languages recognized by a non-deterministic RAMs in quasilinear time.
