  We broadly generalise Mermin-type arguments on GHZ states, and we provide
exact group-theoretic conditions for non-locality to be achieved. Our results
are of interest in quantum foundations, where they yield a new hierarchy of
quantum-realisable All-vs-Nothing arguments. They are also of interest to
quantum protocols, where they find immediate application to a non-trivial
extension of the hybrid quantum-classical secret sharing scheme of Hillery,
Bu\v{z}ek and Berthiaume (HBB). Our proofs are carried out in the graphical
language of string diagrams for dagger compact categories, and their validity
extends beyond quantum theory to any theory featuring the relevant algebraic
structures.
