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The search for sharp constants for inequalities of the type Littlewood's 4/3and Bohnenblust-Hille, besides its pure mathematical interest, has shownunexpected applications in many different fields, such as Analytic NumberTheory, Quantum Information Theory, or (for instance) in deep results on the$n$-dimensional Bohr radius. The recent estimates obtained for the multilinearBohnenblust-Hille inequality (in the case of real scalars) have been recentlyused, as a crucial step, by A. Montanaro in order to solve problems in thetheory of quantum XOR games. Here, among other results, we obtain new upperbounds for the Bohnenblust-Hille constants in the case of complex scalars. Forbilinear forms, we obtain the optimal constants of variants of Littlewood's 4/3inequality (in the case of real scalars) when the exponent 4/3 is replaced byany $r\geq4/3.$ As a consequence of our estimates we show that the optimalconstants for the real case are always strictly greater than the constants forthe complex case.

On the Bohnenblust-Hille inequality and a variant of Littlewood's 4/3 inequality