  We introduce the measures, Bell discord (BD) and Mermin discord (MD), to
characterize bipartite quantum correlations in the context of nonsignaling (NS)
polytopes. These measures divide the full NS polytope into four regions
depending on whether BD and/or MD is zero. This division of the NS polytope
allows us to obtain a 3-decomposition that any bipartite box with two binary
inputs and two binary outputs can be decomposed into Popescu-Rohrlich (PR) box,
a maximally local box, and a local box with BD and MD equal to zero. BD and MD
quantify two types of nonclassicality of correlations arising from all quantum
correlated states which are neither classical-quantum states nor
quantum-classical states. BD and MD serve us the semi-device-independent
witnesses of nonclassicality of local boxes in that nonzero value of these
measures imply incompatible measurements and nonzero quantum discord only when
the dimension of the measured states is fixed. The 3-decomposition serves us to
isolate the origin of the two types of nonclassicality into a PR-box and a
maximally local box which is related to EPR-steering, respectively. We consider
a quantum polytope that has an overlap with all the four regions of the full NS
polytope to figure out the constraints of quantum correlations.
