# complexity theory – A Query regarding Polynomial hierarchy collapse to a finite level

Assuming a hypothetical scenario that complexity class $$PSPACE$$ is shown to belong to complexity class $$NP^{coNP}$$ the Polynomial Hirerchy Collapses to a finite level.

Query 1: Which level the above complexity class belong to (level 2 or 3)?

Only a handful of important classes will survive this collapse. The only ones I can think of are:
$$NC$$, $$P$$, $$MA$$, $$AM$$, $$NP$$, $$coNP$$, $$SZK$$, $$BPP$$, $$PH$$, $$P/Poly$$, $$NP/poly$$

Query 2: Can someone enumerate some other important classes we missed?

Query 3: Now, given this unlikely result, what other problems still remain open in the complexity hierarchy above $$P$$ and below $$EXP$$ (other that the three below):

1. $$P$$ vs $$NP$$
2. $$NP$$ vs $$coNP$$
3. $$P$$ vs $$BPP$$

Query 4: Does the above result also prove $$AM=MA$$?

In general this query is regarding the general landscape of Complexity Hierarchy in this unlikely scenario.