Recently I was asking about the impact of the groundbreaking result MIP*=RE on logic and proof theory (see this discussion). Surprising as it is I got confused with the following: MIP* is a ,,quantum'' version of MIP where two provers are allowed to use only *classical correlations*. My intuition was that allowing more general correlations would *always* make provers more powerful: it turned out to be the case for quantum correlation but here it is claimed (somewhere in the discussion) that it was believed that MIP* would turn out to be *smaller* than MIP. And in fact allowing even stronger correlations, the so called *non-signalling* correlation we get another class $MIP_{ns}$ which turns out to be *smaller* than MIP: namely $MIP_{ns}=EXP$ while $MIP=NEXP$.

What is the intuitive reason behind this fact? Why allowing more general correlations won't give yet another bigger complexity class?

do knowthis. They can decide either option: to convince me rightlyorto trick me (they can decide to trick me in both cases: if the machine eventually halts or if the machine will never halt). If they will always be honest and I will know that they will be then increasing the set of correlations (i.e. increasing their communication... $\endgroup$1more comment