  We investigate the optimality problem associated with the best constants in a
class of Bohnenblust--Hille type inequalities for $m$--linear forms. While
germinal estimates indicated an exponential growth, in this work we provide
strong evidences to the conjecture that the sharp constants in the classical
Bohnenblust--Hille inequality are universally bounded, irrespectively of the
value of $m$; hereafter referred as the \textit{Universality Conjecture}. In
our approach, we introduce the {notions of entropy and complexity}, designed to
measure, to some extent, the complexity of such optimization problems. We show
that the notion of entropy is critically connected to the Universality
Conjecture; for instance, that if the entropy grows at most exponentially with
respect to $m$, then the optimal constants of the $m$% --linear
Bohnenblust--Hille inequality for real scalars are indeed bounded universally
in $m$. It is likely that indeed the entropy grows as $4^{m-1}$, and in this
scenario, we show that the optimal constants are precisely $2^{1-\frac{1}{m}}
$. In the bilinear case, $m=2$, we show that any extremum of the Littlewood's
$4/3$-inequality has entropy $4$ and complexity $2$, and thus we are able to
classify all extrema of the problem. We also prove that, for any mixed $\left(
\ell _{1},\ell _{2}\right) $% --Littlewood inequality, the entropy do grow
exponentially and the sharp constants for such a class of inequalities are
precisely $(\sqrt{2})^{m-1}$. In addition to the {notions of entropy and
complexity}, the approach we develop in this work makes decisive use of a
family of strongly non-symmetric $m$--linear forms, which has further
consequences to the theory, as we explain herein.
