On the Everett interpretation of quantum mechanics, the wavefunction is the whole truth about the physical world. Moreover, the wavefunction never collapses, and we live in a vast multiverse. On functionalism about mind, mental properties supervene on functional properties of the world. We can specify this further: it could be that all mental properties supervene on functional properties of the world, or just the non-qualitative ones or just the narrow-content non-qualitative ones. My argument will work in all these cases.
I claim that the Everett interpretation and functionalism about mind are not both true.
The argument is fairly simple. Any two worlds that are isomorphic under an isomorphism of the quantum structure (i.e., of the Hilbert spaces and the operator algebras) have the same functional properties. Now consider two worlds w1 and w2. Both are short-lived worlds: the temporal sequence of each is only a billion years old. Each world is an exact duplicate of a temporal portion of our world. Thus, w1 is an exact duplicate of the temporal portion of our world from 13 billion years ago to 12 billion years ago, while w2 is an exact duplicate of the temporal portion of our world from a billion years ago to the present. Then w2 has the same kind of mental properties that obtained in our world over the last billion years. And w1 has the same kind of mental properties that obtained in our world from 13 to 12 billion years ago.
But there is a quantum-structure preserving isomorphism from w1 to w2. This isomorphism is simply given by the time-evolution operator U12 (where we measure time in billions of years). This operator is an isomorphism of the quantum structure. Hence w1 and w2 are exactly alike with respect to mental properties. Hence our world had exactly the same mental properties in the early 13-to-12 billion-years-ago period as in the last billion years. That's absurd. (For one, it makes us question how we could possibly know that the world is as old as we think it is.)
Generalizing, it seems that a functionalist Everettian picture gives rise to a world where mental properties don't change: every interval of time long enough to include mental properties has the same mental properties as any other interval of equal length.
Objection: One might think that mental properties depend on evolutionary history. Taking a billion year segment is enough to include all our mentally relevant evolutionary history, so we still get the conclusion that mental states just like ours occurred in the much earlier universe. But we may not get the general conclusion that any two intervals of time have the same mental properties. In particular, an evolutionary functionalist might think that a world w2* that includes only the last 100 years of our universe's physical history wouldn't actually have any mental properties because it wouldn't include enough evolutionary history.
Response: Maybe. But we still have enough to argue for the absurd conclusion that we can't tell whether we're approximately 13.8 billion years from the beginning of the universe or approximately 1.8 billion years from it.
Objection: The Everettian should say: "I already knew this. It was very unlikely for life to evolve within the first 1.8 billion years, but it wasn't impossible, and since all possibilities are realized in the multiverse, that one was, too."
Response: If that's right, then the argument applies not just against the functionalist Everettian but against every Everettian.