# Survivors classes in the extremely unlikely scenario of P = PSPACE

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:

1. Should be above $$P$$ (thus $$L, NC, NL$$ etc. could be temporarily ignored)
2. 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.

• anyone please ? – TheoryQuest1 Nov 23 '18 at 11:38

The time hierarchy theorem still tells us that $$\mathrm{P}\subsetneq \mathrm{EXP}\subsetneq\mathrm{2\text{-}EXP} \subsetneq \dots\,,$$ and likewise for the analogous space classes and nondeterministic time and space classes.