# Importance of $\Sigma^P_2=PSPACE$

Assuming (the unlikely scenario) that the Polynomial Hierarchy collapses to second level. Thus, $$\Sigma_2^p=PSPACE$$.

How significant this result would be (broad consensus) as compared to the other open problems in TCS (the most important two being $$P$$ vs $$NP$$ and $$NP$$ vs $$coNP$$).

Most of the sources rightly point to the above two problems being the most important but despite most of the classes in complexity hierarchy falling under $$PSPACE$$, I am somewhat surprised not much has been discussed about it.

Is it something that is at the Turing Award or Breakthrough Prize level?

Since $$\Sigma^{P}_2 \subseteq PH \subseteq PSPACE$$, then showing this will prove that $$PH=PSPACE$$, showing, for example, this important implication.

In addition, another immediate but important implication is that the polynomial hirarchy collapses to $$\Sigma^P_2$$ - which shows that there is no gain in "strength" by adding more alternations above only 3 alternations (using the definition of $$\Sigma^P_k$$ as an alternating turing machine).

As $$IP=PSPACE$$, this might lead for some more room of thought whether the same statement can be said about interactive protocols - can it be said there exists some $$r$$ such that all of IP is contained in Arthur-Merlin protocols with only $$r$$ rounds (basically, is IP collapsing to AM with $$r$$ rounds)?

Therefore, yes - it definitely would be considered a big breakthrough.