Nontrivial separation consequences of P!=NP I am looking for nontrivial examples of complexity class separations that are known to follow from the P!=NP hypothesis. By a "nontrivial example" I mean that it is not just an automatic consequence of an easy containment relationship, such as P!=PSPACE, P!=PH etc. would be.
De Loera, Lee,  Malkin, and Margulies developed an algorithm for determining infeasibility of certain combinatorial problems using Hilbert's Nullstellensatz.

For example, they take the constraints of 3-coloring a graph and describe that problem as a system of polynomial equations.  They compute a set of certificate polynomials which verify that these polynomials have no solutions (using the Nullstellensatz).  To make this computation finite, they bound the degree of the certificates.  For the 3-coloring case, they were unable to find a non-3-colorable graph which required degree higher than four for the certificate.

If P $\neq$ NP, then P $\neq$ coNP and the degree of a Nullstellensatz Certificate for this problem is unbounded, as a finite bound would yield a polynomial-time algorithm.