# For all oracles A, If $P^A \neq PSPACE^A$, then Does it imply that $P \neq PSPACE$?

My question related to relativized world. I would like to know about how to show that class is different from another class in the oracle world and whether this applied to our real world.

For example, let us have the following assumption: suppose for all oracles A we have $$P^A \neq PSPACE^A$$, then Does it imply that $$P = PSPACE$$?

• Consider an "useless" oracle (say, $A = \emptyset$). Jan 19 '21 at 18:33

If $$A = \emptyset$$, or more generally, if $$A \in \mathsf{P}$$, then $$\mathsf{P}^A = \mathsf{P}$$ and $$\mathsf{PSPACE}^A = \mathsf{PSPACE}$$, and so $$\mathsf{P}^A \neq \mathsf{PSPACE}^A$$ is the same as $$\mathsf{P} \neq \mathsf{PSPACE}$$.
It is also interesting to ask whether $$\mathsf{P}^A \neq \mathsf{PSPACE}^A$$ for a random oracle implies $$\mathsf{P} \neq \mathsf{PSPACE}$$. This was disproved (with $$\mathsf{P}$$ replaced by $$\mathsf{IP}$$) by Chang, Chor, Goldreich, Hartmanis, Håstad, Ranjan, and Rohatgi in their paper The random oracle hypothesis is false.