What role has the signature of the metric tensor in Relativity in the fact that instead of using classical probability theory in quantum mechanics, we use complex amplitudes that need to be squared?
Quantum Mechanics and Quantum Field Theory are successful theories because they provide a very precise description of our observations. However their foundations is based on Probability Theory where wave function evolves according to Schrodinger equation until it collapses into one or another outcome (like spin up/down). 
Replacing Classical probabilities by Complex Amplitudes allows us  to get rid off diverging terms which arise from factor ordering problem but still requires non-locality which seems odd at first sight as these theories were meant to describe particles far apart. It leads however both schools of thought(QM + QFT) eventually: QFT gets an extra bonus of being relativistically invariant allowing you to write down interaction Lagrangian in terms only of known interactions among fundamental particle family plus photons for instance. The rest remains valid. Meanwhile both theories becomes very complicated when expressed with traditional mathematical machinery available to all students of physics like Calculus or Linear Algebras since no longer everything comes free after a single step anymore