Edit: As Ravi Boppana correctly pointed out in his answer and Scott Aaronson also added another example in his answer, the answer to this question turned out to be “yes” in a way which I had not expected at all. First I thought that they did not answer the question I had wanted to ask, but after some thinking, these constructions answer at least one of the questions I wanted to ask, that is, “Is there any way to prove a conditional result ‘P=NP ⇒ L∈P’ without proving the unconditional result L∈PH?” Thanks, Ravi and Scott!
Is there a decision problem L such that the following conditions are both satisfied?
- L is not known to be in the polynomial hierarchy.
- It is known that P=NP will imply L∈P.
An artificial example is as good as a natural one. Also, although I use the letter “L,” it can be a promise problem instead of a language if it helps.
Background. If we know that a decision problem L is in the polynomial hierarchy, then we know that “P=NP ⇒ L∈P.” The intent of the question is to ask whether the converse holds. If a language L satisfying the above two conditions exists, then it can be thought of as an evidence that the converse fails.