The fact that first-person conscious perceptions contain many bits of information strongly suggests they are produced nonlocally by the effects of many atoms, possibly via nonlocal quantum operators. If these operators act back on the quantum state of atoms, they could cause signals to propagate faster than light, violating the usual causality of local quantum field theory. Although there is no direct evidence yet that such nonlocal operators associated with psycho-physical parallelism act back on the quantum state, it is not totally implausible. In principle, the resulting superluminal propagation might be observable by sending signals across brain regions that lead to conscious perceptions.
Sensible Quantum Mechanics (also called Mindless Sensationalism) is a framework that connects conscious experience to a quantum universe by assigning each perception a measure equal to the expectation value of a corresponding quantum "awareness operator" in a fixed universal quantum state. These measures act like probabilities for a vast set of perceptions that all exist with varying degrees of reality, making detailed theories testable. The framework is deterministic—there are no propensities or indeterministic actualizations—and thus no incompatibilistic free will. Conscious perceptions are epiphenomena, determined by the awareness operators and the quantum state. Unlike "many-minds" interpretations, each perception is an entire conscious experience with no direct contact with others, making this a "many-perceptions" variant of Everett's many-worlds framework.
A framework called Sensible Quantum Mechanics proposes a precise parallel between conscious experiences and the mathematical structures of quantum physics. Conscious experiences correspond to quantum operators, and the measure of a set of possible experiences is given by the expectation value of a corresponding operator (a positive-operator-valued measure). This leads to the Conditional Aesthemic Principle: given that we are conscious beings, our conscious experiences are likely to be typical among all conscious experiences with that measure. The ideas are expressed in lay terms for philosophers of consciousness and others.