Oracular separations between poly- and log-depth quantum circuits The following problem appears in Aaronson's list Ten Semi-Grand Challenges for Quantum Computing Theory.


  Is $\mathsf{BQP}=\mathsf{BPP}^{\mathsf{BQNC}}$ In other words, can the "quantum" part of any quantum algorithm be compressed to $\mathrm{polylog}(n)$ depth, provided we're
  willing to do polynomial-time classical postprocessing? (This is known
  to be true for Shor's algorithm.) If so, building a general-purpose
  quantum computer would be much easier than is generally believed!
  Incidentally, it's not hard to give an oracle separation between $\mathsf{BQP}$
  and $\mathsf{BPP}^{\mathsf{BQNC}}$, but the question is whether there's any concrete function "instantiating" such an oracle.


It has been conjectured by Jozsa that the answer to the question is yes in the ''measurement-based model of quantum computation": where local measurements, adaptive local gates and efficient classical post-processing are allowed. See also this related post.

Question. I would like to know about the currently-known oracular separations between this classes (or, at least, the oracle separation to which  Aaronson is referring to).
I apologize; I was too glib when I wrote that.  While I believe it's possible to prove an oracle separation between $BQP$ and $BPP^{BQNC}$ using current techniques, it hasn't been done (12 years after I first thought about the problem, then put it off!), and would certainly be worth a paper for whoever did it.  Maybe your post will help motivate me to finally kill this problem off!