  We provide the two fundamental sets of functional relations which describe
the strong coupling limit of scattering amplitudes in $\mathcal{N} = 4$ SYM
dual to Wilson loops in $AdS_3$: the basic $QQ$-system and the derived
$TQ$-system. We use the $TQ$ relations and the knowledge of the main properties
of the $Q$-function (eigenvalue of some $Q$-operator) to write the Bethe Ansatz
equations, {\it viz.} a set of ('complex') non-linear-integral equations, whose
solutions give exact values to the strong coupling amplitudes/Wilson loops.
Moreover, they have some advantages with respect to the ('real')
non-linear-integral equations of Thermodynamic Bethe Ansatz and still
reproduce, both analytically and numerically, the findings coming from the
latter. In any case, these new functional and integral equations give a larger
perspective on the topic.
