  Quantum contextuality is a source of quantum computational power and a
theoretical delimiter between classical and quantum structures. It has been
substantiated by numerous experiments and prompted generation of state
independent contextual sets, that is, sets of quantum observables capable of
revealing quantum contextuality for any quantum state of a given dimension.
There are two major classes of state-independent contextual sets - the
Kochen-Specker ones and the operator-based ones. In this paper, we present a
third, hypergraph-based class of contextual sets. Hypergraph inequalities serve
as a measure of contextuality. We limit ourselves to qutrits and obtain
thousands of 3-dim contextual sets. The simplest of them involves only 5
quantum observables, thus enabling a straightforward implementation. They also
enable establishing new entropic contextualities.
