  The aim of this paper is to indicate possible applications of operator
systems in qualitative description of varoius scenarios while studying
non-locality. To this end we study in details the notion of generalized
non-commuting cube. Following ideas of Fritz and
Farenick-Kavruk-Paulsen-Todorov we show in systematic way that various classes
of Tsirelson's correlation boxes as well as NPA hierarchies can be described by
using various operator system tensor products of generalized non-commuting
cubes. Moreover, we show also that noncommuting cubes can be applied for the
description of steering assemblages. Next we study some aproximation properties
of noncommuting cubes by finite dimensional models. Finaly, we indicate
possibility to use the framework operator systems for studying Bell and
steering inequalities.
