If the exponential time hypothesis is correct, then one can not solve 3SAT for instances with polyglog number of variables in polynomial time. Of course, quasi-polynomial holds readily.

If the exponential time hypothesis is correct (or even weaker versions), then one can not solve 3SAT for instances with polyglog number of variables in polynomial time. Of course, quasi-polynomial time can solve such instances readily.

While we know that there must be problems in time class $T(n ) * \log n$ which is not in $T(n)$, for any $T(n)$, this is not a useful natural problem (this is a standard result in complexity). In any case, finding a problem that is in QP but not in P would be a huge result. We currently dont even know of natural problems in NP that require more than, say, quadratic time in the general RAM model. Because lower bounds are really really really hard. Thus, the resort to the ETH, unique game conjecture, praying, and proving that problems are NP-Complete.