Does there exist any unrelativized separation between a quantum complexity class and a classical one?
Separations are known for constant-depth circuits. One recent result in this area is due to Bravyi, Gosset, and Koenig (see also the journal version in Science). Recent independent works by Bene Watts, Kothari, Schaeffer, and Tal, Coudron, Stark and Vidick, and Le Gall strengthened the hardness result for classical circuits from worst-case hardness to average-case hardness.