QHk is defined to be PNP[k]; that is, P with k queries to an NP oracle (where k is a constant). Then QH is the union of QHk over all nonnegative k.
QH = BH [Wag88]; thus, either both hierarchies are infinite or both collapse to some finite level.
The smallest class that contains NP and is closed under union, intersection, and complement.
The levels are defined as follows:
Then BH is the union over all i of BHi.
Defined in [WW85].
For more detail see [CGH+88].
Contained in Δ2P and indeed in PNP[log].
If BH collapses at any level, then PH collapses to Σ3P [Kad88].