Is $PSPACE$ believed to be different than $PP$?

Question

From Googling, I couldn't find any discussion about whether $PP=PSPACE$ is more or less likely than $PP\subsetneq PSPACE$.

Is it currently believed that $PP\neq PSPACE$?

What would be the implications if the two happen to be equal?

Answer 1

I hope someone with more knowledge can supply an additional answer.

I don't have a reference or a survey*, but in my experience people expect that $\text{PP}\subsetneq \text{PSPACE}$, mostly because, by default, complexity classes that "look" unequal are usually assumed to be unequal unless there is a very good reason to think otherwise; and because this heuristic has proved useful more often than not.

There is not much evidence for either equality or inequality. A weak form of evidence is that $\text{PP}^{O}\subsetneq \text{PSPACE}^{O}$ relative to a random oracle $O$. Once upon a time, the Random Oracle Hypothesis conjectured that such a random oracle separation would imply a separation in the real world, but this was overturned when $\text{IP}=\text{PSPACE}$ was proved (which, incidentally, also overturned the assumption above, that different-looking complexity classes are always different).

The class $\text{PP}$ can be generalized to the counting hierarchy, $\text{CH}=\text{PP}\cup \text{PP}^{\text{PP}}\cup \text{PP}^{\text{PP}^{\text{PP}}}\cup\cdots$. If $\text{PP}=\text{PSPACE}$ then also $\text{CH}=\text{PSPACE}$. So, a weak form of evidence that $\text{PP}\subsetneq \text{PSPACE}$ would be to find an oracle relative to which $\text{CH}\subsetneq \text{PSPACE}$, but such an oracle has not yet been found. A good start would be to address the second level of the counting hierarchy and find an oracle relative to which $\text{PP}^{\text{PP}}\subsetneq \text{PSPACE}$, but we don't have this either.

*We have a survey with $n=1$ person, namely Ryan Williams [1] gives a 50% probability to $\text{TC}^0=\text{NC}^1$, and provides some reasons. Since $\text{TC}^0=\text{NC}^1$ implies $\text{PP}=\text{PSPACE}$, logically he gives at least 50% probability to $\text{PP}=\text{PSPACE}$. (thanks to domotorp for finding this resource) An implication in the other direction is not known.

[1] Ryan Williams. Some Estimated Likelihoods For Computational Complexity. https://people.csail.mit.edu/rrw/likelihoods.pdf