  Finding a set of empirical criteria fulfilled by any theory that satisfies
the generalized notion of noncontextuality is a challenging task of both
operational and foundational importance. The conventional approach of deriving
facet inequalities from the relevant noncontextual polytope is computationally
demanding. Specifically, the noncontextual polytope is a product of two
polytopes, one for preparations and the other for measurements, and the
dimension of the former typically increases polynomially with the number of
measurements. This work presents an alternative methodology for constructing a
polytope that encompasses the actual noncontextual polytope while ensuring that
the dimension of the polytope associated with the preparations remains constant
regardless of the number of measurements and their outcome size. In particular,
the facet inequalities of this polytope serve as necessary conditions for
noncontextuality. To demonstrate the efficacy of our methodology, we apply it
to nine distinct contextuality scenarios involving four to nine preparations
and two to three measurements to obtain the respective sets of facet
inequalities. Additionally, we retrieve the maximum quantum violations of these
inequalities. Our investigation uncovers many novel non-trivial
noncontextuality inequalities and reveals intriguing aspects and applications
of quantum contextual correlations.
