  Quantum contextuality turns out to be a necessary resource for universal
quantum computation and also has applications in quantum communication. Thus it
becomes important to generate contextual sets of arbitrary structure and
complexity to enable a variety of implementations. In recent years, such
generation has been done for contextual sets known as Kochen-Specker sets. Up
to now, two approaches have been used for massive generation of non-isomorphic
Kochen-Specker sets: exhaustive generation up to a given size and downward
generation from master sets and their associated coordinatizations. Master sets
were obtained earlier from serendipitous or intuitive connections with
polytopes or Pauli operators, and more recently from arbitrary vector
components using an algorithm that generates orthogonal vector groupings from
them. However, both upward and downward generation face an inherent exponential
complexity barrier. In contrast, in this paper we present methods and
algorithms that we apply to downward generation that can overcome the
exponential barrier in many cases of interest. These involve tailoring and
manipulating Kochen-Specker master sets obtained from a small number of simple
vector components, filtered by the features of the sets we aim to obtain. Some
of the classes of Kochen-Specker sets we generate contain all previously known
ones, and others are completely novel. We provide examples of both kinds in 4-
and 6-dim Hilbert spaces. We also give a brief introduction for a wider
audience and a novice reader.
