A 2 qubit quantum computer can produce EPR correlations. It would take a lot more than 2 classical bits to do the same thing. And whether you think the entire world is just being simulated on a big classical machine does make a difference - the same sort of difference it makes if you prefer solipsism to realism.
It is possible that the entire world is an elaborately rigged classical simulation but such a theory doesn't solve any problems and in fact creates a problem, namely why is the computer running such a simulation?
There is a sense in which simple arithmetical truth, which is so rich that it has
been shown to be non axiomatizable (Godel), "run" all classical computations.
Then the fact that [if we are digitalizable entities we cannot know in which
computations we are---and even that we belong to all computations going
through our possible states---] would explain why our environment, defined
by the collection of our most probable computational histories, appears
to be quantum like.
(Qualitatively we get quickly many consistent histories, and arithmetically
we get, not so quickly (alas), an intriguing sort of quantum logic).
Schmidhuber gives another explanation in term of universal prior, but he
does not explain how we would remain (with high probability) in the particular
computation originating from that universal prior.
We cannot simulate in real time and real space a quantum computer
because the whole of physics emerges from a continuum of classical
computational histories. More precisely we cannot simulate anything
describing the result of the observations of whatever exists below our common
substitution level (the level of description such that we would survive a
functional digital substitution made at that level).