In Descriptive Complexity, Immerman has
Corollary 7.23. The following conditions are equivalent:
1. P = NP.
2. Over finite, ordered structures, FO(LFP) = SO.
This can be thought of as "amplifying" P=NP to an equivalent statement over (presumably) larger complexity classes. Note that SO captures the polynomial-time hierarchy PH, and that FO(LFP) captures P, so this can be thought of as P=NP iff P=PH.
(The interesting part of this is the statement that P=NP implies P=PH; it is trivial that P=CC implies P=NP for any class CC that contains NP. Immerman simply remarks "if P=NP then PH=NP", presumably because P=NP can be used with the oracle definition of PH to show inductively that the whole hierarchy collapses.)
My question is:
How much further can P=NP be amplified in this way?
In particular, what is the largest known class CC' such that P=NP implies P=CC', and the smallest class CC such that P=NP implies CC=NP? This would allow P=NP to be replaced by the equivalent question CC=CC'. P appears to be a rather powerful class, which seems to provide little "wiggle room" for arguments trying to separate it from NP: how far can the wiggle room be amplified?
I would of course also be interested in an argument that shows that P=PH is the limit of this approach.
Edit: note the closely related question Why doesn't P=NP imply P=AP (i.e. P=PSPACE)? which focuses on the other direction, why we don't have proofs that P=PSPACE. Answers there by Kaveh and Peter Shor argue that the number of alternations being fixed is key. Another related question is A decision problem which is not known to be in PH but will be in P if P=NP which asks for a candidate problem; the answers there also can be used to construct answers for this question, although these classes are somewhat artificial (thanks to Tsuyoshi Ito for pointing this out). In a more general setting, Collapsing of exptime and alternation bounded turing machine asks whether a local collapse at any level in an alternation hierarchy induces an upward collapse, as happens with the polynomial-time hierarchy.