  Motivated by recent observation of a new high field phase near saturation in
chromium spinels $A$Cr$_2$O$_4$ ($A=$ Zn, Cd, Hg), we study the $S = 3/2$
pyrochlore Heisenberg antiferromagnet with biquadratic interactions. Magnon
instability analysis at the saturation field reveals that a very small
biquadratic interaction can induce magnon pairing in pyrochlore
antiferromagnets, which leads to the emergence of a ferro-quadrupolar phase, or
equivalently a spin nematic phase, below the saturation field. We present the
magnetic phase diagram in an applied field, studying both $S=3/2$ and $S=1$
spin systems. The relevance of our result to chromium spinels is discussed.
