That is, is there a problem that all problems in the polynomial hierarchy can be reduced to in polynomial time, but which some PSPACE problem cannot be reduced to in polynomial time? Clearly if the polynomial hierarchy collapses, then there is such a problem if and only if $PH\neq PSPACE$. Is it known whether there is such a problem, or are there reasons for suspecting one way or the other, assuming the polynomial hierarchy does not collapse?



Should we expect there to be a problem that is PH-hard but not PSPACE-hard?
Alexander Shen suggests the following language that is PH-hard,


but does not seem to be PSPACE-hard: 


STRANGE TQBF $f \in L \Leftrightarrow \forall y\_1 \exists y\_2 \exists y\_3 \forall y\_4 \forall y\_5 \forall y\_6 \forall y\_7 \ldots f(y\_1, y\_2, \ldots, y\_n) =1$.


(So there are $O(\log n)$ changes of quantifiers, $f$ is a boolean formula.)


