It's been a while since I studied this, and I know that some Copenhagenist try to wiggle their way out of this, but if you consider the probability waves in the Copenhagen interpretation to be "real" and "measurement" to cause their collapse, book-keeping information by the universe would seem to have to travel at faster than the speed of light, IIRC.

On the other hand, it's certainly been proven that this cannot be used to transmit information that you might like to at faster than the speed of light. So causality is preserved in any case.