  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.
