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