  The classical Cuntz semigroup has an important role in the study of
C*-algebras, being one of the main invariants used to classify recalcitrant
C*-algebras up to isomorphism. We consider C*-algebras that have Hopf algebra
structure, and find additional structure in their Cuntz semigroups, thus
generalizing the equivariant Cuntz semigroup. We develop various aspects of the
theory of such semigroups, and in particular, we give general results allowing
classification results of the Elliott classification program to be extended to
classification results for C*-algebraic quantum groups.
