  We theoretically consider the superconductivity of the topological
half-Heusler semimetals YPtBi and LuPtBi. We show that pairing occurs between
j=3/2 fermion states, which leads to qualitative differences from the
conventional theory of pairing between j=1/2 states. In particular, this
permits Cooper pairs with quintet or septet total angular momentum, in addition
to the usual singlet and triplet states. Purely on-site interactions can
generate s-wave quintet time-reversal symmetry-breaking states with
topologically nontrivial point or line nodes. These local s-wave quintet pairs
reveal themselves as d-wave states in momentum space. Furthermore, due to the
broken inversion symmetry in these materials, the s-wave singlet state can mix
with a p-wave septet state, again with topologically-stable line nodes. Our
analysis lays the foundation for understanding the unconventional
superconductivity of the half-Heuslers.
