The construction, with the tensor product technique, of the higher dimensional quadrature formulas has been developed, among others, by \textit{S. A. Smolyak} [Dokl. Acad. Nauk SSSR 148, 1042-1045 (1963; Zbl 0202.39901)]. Such a formula is obtained recursively from a sequence of 1-dimensional quadrature rules, for continuous functions in \( C([0, 1]). \) A possible measure of the precision of multivariate Smolyak rules is the \( L_2\)-discrepancy. The authors obtain a recursive algorithm for computing the \( L_2\)-discrepancy of general \( d\)-dimensional Smolyak quadratures with \( N \) nodes. Under some assumptions it is proved that the complexity of this algorithm is \[ O[N(\log N)^{2-d} + d (\log N)^4]. \] Numerical experiments, tables and figures are also presented. The authors compare the \( L_2\)-discrepancy of Smolyak quadrature rules with the square mean of the \( L_2\)-discrepancy of Monte-Carlo integration. So, in the aforementioned comparison, is treated the case when the starting sequences of 1-dimensional quadrature rules for the Smolyak formula are the trapezoidal rules, Newton-Cotes formulas of degree 4 and Clenshaw-Curtis formulas.