Classical Calogero–Moser models with rational potential are known to be superintegrable. That is, on top of the r involutive conserved quantities necessary for the integrability of a system with r degrees of freedom, they possess an additional set of r−1 algebraically and functionally independent globally defined conserved quantities. At the quantum level, Kuznetsov uncovered the existence of a quadratic algebrastructure as an underlying key for superintegrability for the models based on A type root systems. Here we demonstrate in a universal way the quadratic algebrastructure for quantum rational Calogero–Moser models based on any root systems.