Back in 2005, Scott Aaronson posted a list of 10 "semi-grand" challenges for quantum computing theory which contained the following challenge:
The power of small-depth quantum circuits. Is $BQP = BPP^{BQNC}$? In other words, can the "quantum" part of any quantum algorithm be compressed to 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 $BQP$ and $BPP^{BQNC}$, but the question is whether there's any concrete function "instantiating" such an oracle.
I am currently searching for a Ph.D. project and this is the kind of stuff that interests me. I would certainly like to contribute to solving this problem, if it is still open. That is why I would like to know some of the recent developments on this. Is this still an open problem? What contributions were made in the last decade?