If FTL signaling were possible, would it break the framework of Quantum Mechanics the same way it breaks Special Relativity?

If we were to hypothetically introduce a mechanism for true FTL signaling (violating the No-Communication Theorem), we know it would break causality in Special Relativity. What would be the consequences for Quantum Mechanics or Quantum Field Theory? Does the mathematical framework of QM inherently rely on a strict speed limit for information, or would it still function? FTL signaling in QM does it also break down?