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We introduce the measures, Bell discord (BD) and Mermin discord (MD), tocharacterize bipartite quantum correlations in the context of nonsignaling (NS)polytopes. These measures divide the full NS polytope into four regionsdepending on whether BD and/or MD is zero. This division of the NS polytopeallows us to obtain a 3-decomposition that any bipartite box with two binaryinputs 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 MDquantify two types of nonclassicality of correlations arising from all quantumcorrelated states which are neither classical-quantum states norquantum-classical states. BD and MD serve us the semi-device-independentwitnesses of nonclassicality of local boxes in that nonzero value of thesemeasures imply incompatible measurements and nonzero quantum discord only whenthe dimension of the measured states is fixed. The 3-decomposition serves us toisolate the origin of the two types of nonclassicality into a PR-box and amaximally local box which is related to EPR-steering, respectively. We considera quantum polytope that has an overlap with all the four regions of the full NSpolytope to figure out the constraints of quantum correlations.

Canonical decomposition of quantum correlations in the framework of generalized nonsignaling theories