Ko [26] and Bruschi [11] independently showed that, in
some relativized world, PSPACE (in fact, ⊕P) contains a set
that is immune to the polynomial hierarchy (PH). In this paper, we
study and settle the question of relativized separations with
immunity for PH and the counting classes PP, ${\rm C\!\!\!\!=\!\!\!P}$, and ⊕P
in all possible pairwise combinations. Our main result is that there
is an oracle A relative to which ${\rm C\!\!\!\!=\!\!\!P}$ contains a set that is immune BPP⊕P.
In particular, this ${\rm C\!\!\!\!=\!\!\!P}^A$ set is immune to PHA and to ⊕PA. Strengthening results of
Torán [48] and Green [18], we also show that, in suitable relativizations,
NP contains a ${\rm C\!\!\!\!=\!\!\!P}$-immune set, and ⊕P contains a PPPH-immune
set. This implies the existence of a ${\rm C\!\!\!\!=\!\!\!P}^B$-simple set for some
oracle B, which extends results of Balcázar et al. [2,4].
Our proof technique requires a circuit lower bound for “exact
counting” that is derived from Razborov's [35] circuit
lower bound for majority.