  Multiloop scattering amplitudes describing the quantum fluctuations at
high-energy scattering processes are the main bottleneck in perturbative
quantum field theory. The loop-tree duality is a novel method aimed at
overcoming this bottleneck by opening the loop amplitudes into trees and
combining them at integrand level with the real-emission matrix elements. In
this Letter, we generalize the loop-tree duality to all orders in the
perturbative expansion by using the complex Lorentz-covariant prescription of
the original one-loop formulation. We introduce a series of mutiloop topologies
with arbitrary internal configurations and derive very compact and factorizable
expressions of their open-to-trees representation in the loop-tree duality
formalism. Furthermore, these expressions are entirely independent at integrand
level of the initial assignments of momentum flows in the Feynman
representation and remarkably free of noncausal singularities. These
properties, that we conjecture to hold to other topologies at all orders,
provide integrand representations of scattering amplitudes that exhibit
manifest causal singular structures and better numerical stability than in
other representations.
