# A query regarding the definition of Polynomial Hierarchy and PSPACE

If a $$\mathsf{PSPACE-Complete}$$ problem (say $$\mathsf{TQBF}$$) has an algorithm that has the runtime $$\mathsf{NP^{NP}}$$ it follows that $$\mathsf{PSPACE=\Sigma^P_2}$$.

Does that also imply $$\mathsf{\Sigma^P_2 = \Pi^p_2}$$? Moreover, does that imply $$\mathsf{\Delta^p_2\subset \Sigma^P_2}$$ or is that question still open?

Since PSPACE is closed under complementation, if $$\mathsf{PSPACE} = \Sigma_2^P$$ then $$\Sigma_2^P = \mathsf{PSPACE} = \mathsf{coPSPACE} = \mathsf{co}\Sigma_2^P = \Pi_2^P.$$ We always have $$\Delta_2^P \subseteq \Sigma_2^P$$. I'm not sure if you can deduce the opposite inclusion under your assumption.
• Thank you. I am unclear about the statement: "..opposite inclusion..". If I am correct it means that it cannot tell if the inclusion $\Delta_2^P \subseteq \Sigma_2^P$ is proper or not? Oct 29 at 14:02
• Given an inclusion $A \subseteq B$, the opposite inclusion is $B \subseteq A$. Oct 29 at 14:03
• I'm not sure how you would deduce $\Delta_2^P \neq \Sigma_2^P$; for all we know, it could be that P equals NP. Oct 29 at 14:03