  This paper studies multiple-proof quantum Merlin-Arthur (QMA) proof systems
in the setting when the completeness-soundness gap is small. Small means that
we only lower-bound the gap with an inverse-exponential function of the input
length, or with an even smaller function. Using the protocol of Blier and Tapp
[arXiv:0709.0738], we show that in this case the proof system has the same
expressive power as non-deterministic exponential time (NEXP). Since
single-proof QMA proof systems, with the same bound on the gap, have expressive
power at most exponential time (EXP), we get a separation between single and
multi-prover proof systems in the 'small-gap setting', under the assumption
that EXP is not equal to NEXP. This implies, among others, the nonexistence of
certain operators called disentanglers (defined by Aaronson et al.
[arXiv:0804.0802]), with good approximation parameters.
  We also show that in this setting the proof system has the same expressive
power if we restrict the verifier to be able to perform only Bell-measurements,
i.e., using a BellQMA verifier. This is not known to hold in the usual setting,
when the gap is bounded by an inverse-polynomial function of the input length.
To show this we use the protocol of Chen and Drucker [arXiv:1011.0716]. The
only caveat here is that we need at least a linear amount of proofs to achieve
the power of NEXP, while in the previous setting two proofs were enough.
  We also study the case when the proof-lengths are only logarithmic in the
input length and observe that in some cases the expressive power decreases.
However, we show that it doesn't decrease further if we make the proof lengths
to be even shorter.
