  We propose novel triple-$q$ multipole orders in quadrupolar Kondo system
PrV$_2$Al$_{20}$. Anisotropies of quadrupole moments stabilize a triple-$q$
order, which further leads to the second transition to a coexisting phase with
triple-$q$ octupole moments. The mechanism for realizing these triple-$q$
states is unique to time-reversal even multipolar systems. There, the cubic
invariant of quadrupole moments with triple-$q$ components in its free energy
plays an important role. We analyze a multipolar exchange model by mean-field
approximation and discuss the temperature and magnetic-field phase diagrams.
Many of the microscopic results such as finite ferro octupole moments in the
coexisting phase and the field-induced phases are qualitatively consistent with
the experiments in PrV$_2$Al$_{20}$.
