If its proven today that $P=PSPACE$ the whole polynomial hierarchy collapses. Moreover since $BQP \subseteq PSPACE$, it will also result in $P=BQP$.
However unlikely the above scenario is, hypothetically if this were the case what are the complexity classes (the most important ones) that will survive this collapse into $P$. The classes we are talking about:
- Should be above $P$ (thus $L, NC, NL$ etc. could be temporarily ignored)
- Are interesting and open enough to be studied (thus $E, EE, ALL$ etc. can be ignored)
We are concerned about both the traditional as well as Quantum Complexity classes.