Oracular separations between poly- and log-depth quantum circuits
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!