  An impressive effort is being placed in order to develop new strategies that
allow an efficient computation of multi-loop multi-leg scattering amplitudes,
with a particular emphasis in removing spurious singularities and
instabilities. In this Letter, we describe an innovative technique to obtain
the causal representation of any multi-loop multi-leg amplitude in Quantum
Field Theory, taking advantage of the fact that all the information about
thresholds is encoded in the cusp structure. By using the Loop-Tree Duality
(LTD) theorem, we first obtain a compact causal representation through the
nested residue computation for several topological families. From their
representation, we extract all the possible combinations of causal propagators,
$\lambda_i^{\pm}$, and then study their geometrical properties. We found that
the causal propagators can be built starting from the cusp matrix, through a
suitable definition of connected partitions. The causal representation for a
given topological family is obtained by summing over all the possible products
of $\lambda_i^{\pm}$ that originate connected and oriented partitions of the
underlying topology. These results are compatible with Cutkowsky rules,
containing only physical thresholds which noticeably improves the numerical
stability. Moreover, we explicitly show that diagrams with the same number of
cusps exhibits similar causal structures, regardless of the number of loops.
