  We study the topological properties of superconductors with paired
$j=\frac{3}{2}$ quasiparticles. Higher spin Fermi surfaces can arise, for
instance, in strongly spin-orbit coupled band-inverted semimetals. Examples
include the Bi-based half-Heusler materials, which have recently been
established as low-temperature and low-carrier density superconductors.
Motivated by this experimental observation, we obtain a comprehensive
symmetry-based classification of topological pairing states in systems with
higher angular momentum Cooper pairing. Our study consists of two main parts.
First, we develop the phenomenological theory of multicomponent (i.e., higher
angular momentum) pairing by classifying the stationary points of the free
energy within a Ginzburg-Landau framework. Based on the symmetry classification
of stationary pairing states, we then derive the symmetry-imposed constraints
on their gap structures. We find that, depending on the symmetry quantum
numbers of the Cooper pairs, different types of topological pairing states can
occur: fully gapped topological superconductors in class DIII, Dirac
superconductors and superconductors hosting Majorana fermions. Notably, we find
a series of nematic fully gapped topological superconductors, as well as
double-Dirac superconductors with quadratic dispersion. Our approach, applied
here to the case of $j=\frac{3}{2}$ Cooper pairing, is rooted in the symmetry
properties of pairing states, and can therefore also be applied to other
systems with higher angular momentum and high-spin pairing. We conclude by
relating our results to experimentally accessible signatures in thermodynamic
and dynamic probes.
