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Quantum contextuality turns out to be a necessary resource for universalquantum computation and important in the field of quantum informationprocessing. It is therefore of interest both for theoretical considerations andfor experimental implementation to find new types and instances of contextualsets and develop methods of their optimal generation. We present an arbitraryexhaustive hypergraph-based generation of the most explored contextualsets---Kochen-Specker (KS) ones---in 4, 6, 8, 16, and 32 dimensions. Weconsider and analyse twelve KS classes and obtain numerous properties oftheirs, which we then compare with the results previously obtained in theliterature. We generate several thousand times more types and instances of KSsets than previously known. All KS sets in three of the classes and in theupper part of a fourth are novel. We make use of the MMP hypergraph language,algorithms, and programs to generate KS sets strictly following theirdefinition from the Kochen-Specker theorem. This approach proves to beparticularly advantageous over the parity-proof-based ones (which prevail inthe literature), since it turns out that only a very few KS sets have a parityproof (in six KS classes 0.01% and in one of them 0%). MMP hypergraph formalismenables a translation of an exponentially complex task of solving systems ofnonlinearequations, describing KS vector orthogonalities, into a statisticallylinearly complex task of evaluating vertex states of hypergraph edges, thusexponentially speeding up the generation of KS sets and enabling us to generatebillions of novel instances of them. The MMP hypergraph notation also enablesus to graphically represent KS sets and to visually discern their features.

Arbitrarily exhaustive hypergraph generation of 4-, 6-, 8-, 16-, and 32-dimensional quantum contextual sets