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Last years, bounds on the maximal quantum violation of general Bellinequalities were intensively discussed in the literature via differentmathematical tools. In the present paper, we analyze quantum violation ofgeneral Bell inequalities via the LqHV (local quasi hidden variable) modellingframework, correctly reproducing the probabilistic description of every quantumcorrelation scenario. The LqHV mathematical framework allows us to derive forall d and N a new upper bound (2d-1)^{N-1} on the maximal violation by anN-qudit state of all general Bell inequalities, also, new upper bounds on themaximal violation by an N-qudit state of general Bell inequalities for Ssettings per site. These new upper bounds essentially improve all the knownprecise upper bounds on quantum violation of general multipartite Bellinequalities. For some S, d and N, the new upper bounds are attainable.

New concise upper bounds on quantum violation of general multipartite Bell inequalities