Ph complete language forces collapse

Given: the objects and hypotheses in the statement above.

For each fixed $k\ge 0$, both ${\mathrm{\Sigma }}_k^p$ and ${\mathrm{\Pi }}_k^p$ are closed under polynomial-time many-one preimages and polynomially many uniformly indexed conjunctions and disjunctions. Precisely, if the language of pairs $(x,i)$ lies in the class and $0\le i<q(\mathrm{\mid }x\mathrm{\mid })$ for a polynomial-time computable polynomial bound $q$, then both “every such $i$” and “some such $i$” define languages in that class. Complements interchange ${\mathrm{\Sigma }}_k^p$ and ${\mathrm{\Pi }}_k^p$, and each is contained in both classes at level $k+1$. These assertions also hold with one fixed base oracle. (Ph quantifier block closure).

For fixed $k\ge 1$, ${\mathrm{\Sigma }}_k^p={\mathrm{\Pi }}_k^p$ implies $\mathrm{P}\mathrm{H}={\mathrm{\Sigma }}_k^p={\mathrm{\Pi }}_k^p$. Also $P=NP$ implies $\mathrm{P}\mathrm{H}=P$. (Sigma k equals pi k implies ph collapse).

TQBF lies in PSPACE, and every language in PSPACE polynomial-time many-one reduces to TQBF. Therefore TQBF is PSPACE-complete. (TQBF is PSPACE-complete).