  In the nonsignaling framework, nonclassicality in correlation arising from
two spatially separated input-output devices gets manifested, solely, through
its \emph{nonlocal} behavior. Study of correlations based on this said feature
is commonly known as local-nonlocal paradigm. While in two-party scenario
correlations can be of only two types either local or nonlocal, the situation
gets more involved for multi-party scenario, \emph{e.g.,} for tripartite
scenario, correlations can be of three types: fully local, two-way local, and
genuinely nonlocal. Nonsignaling correlations having quantum realization are
termed physical. Fully local and certain quantum realizable two-way local
tripartite correlations always have a quantum realization with tripartite
biseparable states if there is no restriction on the local Hilbert-space
dimensions. In this work, we study the quantum simulation of fully local and
two-way local tripartite correlations with restricted local Hilbert-space
dimensions, in particular we consider
$\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2$ quantum systems.
Interestingly, in this restricted simulation scenario we find that simulation
of certain fully local and two-way local correlations necessarily requires
\emph{genuine quantumness} in the three qubit states. This, going beyond the
standard nonlocality paradigm, captures a new notion of genuine nonclassicality
even in the fully local and two-way local correlations. To explore this newly
introduced notion of genuine nonclassicality, we propose two quantities of
interest, called \emph{Svetlichny strength} and \emph{Mermin strength}, and
extensively study their properties.
