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