Non-uniform version for the whole polynomial hierarchy
Well, sure, we know things. I think this is a pretty standard nomenclature for it. This hierarchy collapses if and only if $\mathsf{PH}$ does, exercise:


For one direction, modify the proof of Karp-Lipton to show that if $\mathsf{NP} \subseteq \mathsf{coNP}/poly$ then $\mathsf{PH}$ collapses, and observe that this result relativizes
For the other direction, see the comments by Kaveh below.