  Topological superconductivity in multiband systems has received much
attention due to a variety of possible exotic superconducting order parameters
as well as non-trivial bulk and surface states. While the impact of coexisting
magnetic order on superconductivity has been studied for many years, such as
ferromagnetic superconductors, the implication of coexisting multipolar order
has not been explored much despite the possibility of multipolar hidden order
in a number of $f$-electron materials. In this work, we investigate topological
properties of multipolar superconductors that may arise when quadrupolar local
moments are coupled to conduction electrons in the multiband Luttinger
semimetal. We show that the multipolar ordering of local moments leads to
various multipolar superconductors with distinct topological properties. We
apply these results to the quadrupolar Kondo semimetal system, PrBi, by
deriving the microscopic multipolar Kondo model and examining the possible
superconducting order parameters. We also discuss how to experimentally probe
the topological nature of the Bogoliubov quasiparticles in distinct multipolar
superconductors via doping and external pressure, especially in the context of
PrBi.
