Importance of $\Sigma^P_2=PSPACE$

Question

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?

Answer 1

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.