We study quantum effects in a spin-3/2 antiferromagnet on the pyrochlore lattice in an external magnetic field, focusing on the vicinity of a plateau in the magnetization at half the saturation value, observed in CdCr2O4 and HgCr2O4. Our theory, based on quantum fluctuations, predicts the existence of a symmetry-broken state on the plateau, even with only nearest-neighbor microscopic exchange. This symmetry-broken state consists of a particular arrangement of spins polarized parallel and antiparallel to the field in a 3:1 ratio on each tetrahedron. It quadruples the lattice unit cell, and reduces the space group from Fd3m to P4(3)32. We also predict that for fields just above the plateau, the low-temperature phase has transverse spin order, describable as a Bose-Einstein condensate of magnons. Other comparisons to and suggestions for experiments are discussed.