  Cup products provide a natural approach to access higher bounded cohomology
groups. We extend vanishing results on cup products of Brooks quasimorphisms of
free groups to cup products of median quasimorphisms, i.e., Brooks-type
quasimorphisms of group actions on CAT(0) cube complexes. In particular, we
obtain such vanishing results for groups acting on trees and for right-angled
Artin groups. Moreover, we outline potential applications of vanishing results
for cup products in bounded cohomology.
