  Recently, quantum contextuality has been proved to be the source of quantum
computation's power. That, together with multiple recent contextual
experiments, prompts improving the methods of generation of contextual sets and
finding their features. The most elaborated contextual sets, which offer
blueprints for contextual experiments and computational gates, are the
Kochen--Specker (KS) sets. In this paper, we show a method of vector generation
that supersedes previous methods. It is implemented by means of algorithms and
programs that generate hypergraphs embodying the Kochen-Specker property and
that are designed to be carried out on supercomputers. We show that vector
component generation of KS hypergraphs exhausts all possible vectors that can
be constructed from chosen vector components, in contrast to previous studies
that used incomplete lists of vectors and therefore missed a majority of
hypergraphs. Consequently, this unified method is far more efficient for
generations of KS sets and their implementation in quantum computation and
quantum communication. Several new KS classes and their features have been
found and are elaborated on in the paper. Greechie diagrams are discussed. A
detailed and complete blueprint of a particular 21-11 KS set with a complex
coordinatization is presented in Appendix A, in contrast to the one from the
published version of this paper where only a few of its states were given.

Quantum Physics