Bennett and Gill (1981) showed that P^A != NP^A != coNP^A for a random
oracle A, with probability 1. We investigate whether this result
extends to individual polynomial-time random oracles. We consider two
notions of random oracles: p-random oracles in the sense of
martingales and resource-bounded measure (Lutz, 1992; Ambos-Spies et
al., 1997), and p-betting-game random oracles using the betting games
generalization of resource-bounded measure (Buhrman et al.,
2000). Every p-betting-game random oracle is also p-random; whether
the two notions are equivalent is an open problem.

(1) We first show that P^A != NP^A for every oracle A that is
p-betting-game random.

Ideally, we would extend (1) to p-random oracles. We show that
answering this either way would imply an unrelativized complexity
class separation:

(2) If P^A != NP^A relative to every p-random oracle A, then BPP != EXP.

(3) If P^A = NP^A relative to some p-random oracle A, then P != PSPACE.

Rossman, Servedio, and Tan (2015) showed that the polynomial-time
hierarchy is infinite relative to a random oracle, solving a
longstanding open problem. We consider whether we can extend (1) to
show that PHA is infinite relative to oracles A that are
p-betting-game random. Showing that PHA separates at even its first
level would also imply an unrelativized complexity class separation:

(4) If NP^A != coNP^A for a p-betting-game measure 1 class of oracles
A, then NP != EXP.

(5) If PH^A is infinite relative to every p-random oracle A, then PH
!= EXP.