  As quantum contextuality proves to be a necessary resource for universal
quantum computation, we present a general method for vector generation of
Kochen-Specker (KS) contextual sets in the form of hypergraphs. The method
supersedes all three previous methods: (i) fortuitous discoveries of smallest
KS sets, (ii) exhaustive upward hypergraph-generation of sets, and (iii) random
downward generation of sets from fortuitously obtained big master sets. In
contrast to previous works, we can generate master sets which contain all
possible KS sets starting with nothing but a few simple vector components. From
them we can readily generate all KS sets obtained in the last half a century
and any specified new KS sets. Herewith we can generate sufficiently large sets
as well as sets with definite required features and structures to enable
varieties of different implementations in quantum computation and
communication.
