  Contextuality is a fundamental manifestation of nonclassicality, indicating
that for certain quantum correlations, sets of jointly measurable variables
cannot be pre-assigned values independently of the measurement context. In this
work, we characterize nonclassical quantum correlation beyond contextuality, in
terms of supernoncontextuality, namely the higher-than-quantum
hidden-variable(HV) dimensionality required to reproduce the given
noncontextual quantum correlations. Thus supernoncontextuality is the
contextuality analogue of superlocality. Specifically, we study the quantum
system of two-qubit states in a scenario composed of five contexts that
demonstrate contextuality in a state-dependent fashion. For this purpose, we
use the framework of boxes, whose behavior is described by a set of
probabilities satisfying the no-disturbance conditions. We observe that
superlocal states must have sufficiently a high HV dimension in order to lead
to a contextual box. On the other hand, a noncontextual superlocal box can be
supernoncontextual, but superlocality is not a necessary condition. For
sublocal (i.e., not superlocal) boxes, supernoncontextuality arises owing to a
different type of simultaneous correlation in two MUBs present in any
nonproduct state and correlations due to nonlocal measurements. From an
operational perspective, the superlocal states that can or cannot be used to
demonstrate contextuality can be distinguished by quantum correlations in two
and three mutually unbiased bases (MUBs) of the discordant correlations in the
states.
