The Monte Carlo methods used in Lattice QCD simulations rely on the rotation of the path integral to Euclidean metric. Unfortunately, the limited knowledge of correlation functions on finite subsets of points prevents a direct analytic continuation to Minkowski signature. In their seminal publication of 1990, Maiani and Testa showed that physical amplitudes away from threshold cannot be directly extracted, ie without an inverse problem, from Euclidean correlators, due to off-shell contaminations. In this presentation, we revisit and extend their original work, and explore the connection with recent developments on the inverse problem in Lattice QCD.