Does the existence of PH-complete problems relativize? The Baker-Gill-Solovay result showed that the P = NP question does not relativize, in the sense that no relativizing proof (insensitive to the presence of an oracle) can possibly settle the P = NP question.

My question is:  Is there a similar result for the question, "Does there exist a PH-complete problem?"  An answer in the negative to this question would imply P != NP; an answer in the affirmative would be unlikely but interesting because it would mean that PH collapses to some level.

I'm not sure, but I suspect that a TQBF oracle would lead PH to be equal to PSPACE, and thus to have a complete problem.  In addition to being uncertain regarding this, I am curious as to whether or not there is an oracle relative to which PH provably does not have a complete problem.

-Philip
Yao showed, in 1985, that there exist oracles relative to which the Polynomial Hierarchy is infinite. Relative to such an oracle, there don't exist PH-complete problems.

Also, you are right that with a TQBF oracle, PH equals PSPACE. In fact, even P = PSPACE in the presence of a TQBF oracle.