We study some logical interrelationships between fundamental properties in (relativistic) quantum theories. An operational no-signalling condition is first introduced in the context of quantum mechanics, where we prove its equivalence to an apparently weaker version restricted to ideal measurements, and to a property of factorization of the evolution unitary operator. We then translate this condition in quantum field theory and prove that it implies both microcausality and the spin-statistics theorem, in the ideal case of pointwise measurements implemented in the projection postulate sense. This provides an argument (often invoked but apparently missing in the literature) to see microcausality as a necessary condition for the compatibility of spacelike separated operations.