>>13619677What they are talking about is not a rejection of causality as it is known today a but revision of what it means at the quantum level. In particular they are talking about microcausality, a consequence of the causality principle.
Microcausality states that for two independent measurements of observables at two points with a space-like separation, the order in which they were measured is irrelevant for a sufficiently small time interval since disturbances cannot propagate faster than light speed. This does not violate causality, in fact it is a direct consequence of causality. What it did reject was the classical idea of causality.
This is what they mean when they say QM destroys (classical) causality; it forced an understanding that causality obeys the speed of light, and that a generator of time translation does not require any time derivatives of the field operators.