  Multi Prover Interactive Proof systems (MIPs)were first presented in a
cryptographic context, but ever since they were used in various fields.
Understanding the power of MIPs in the quantum context raises many open
problems, as there are several interesting models to consider. For example, one
can study the question when the provers share entanglement or not, and the
communication between the verifier and the provers is quantum or classical.
While there are several partial results on the subject, so far no one presented
an efficient scheme for recognizing NEXP (or NP with logarithmic
communication), except for [KM03], in the case there is no entanglement (and of
course no communication between the provers). We introduce another variant of
Quantum MIP, where the provers do not share entanglement, the communication
between the verifier and the provers is quantum, but the provers are unlimited
in the classical communication between them. At first, this model may seem very
weak, as provers who exchange information seem to be equivalent in power to a
simple prover. This in fact is not the case - we show that any language in NEXP
can be recognized in this model efficiently, with just two provers and two
rounds of communication, with a constant completeness-soundness gap.
