Interestingly, we'll never have any evidence either way. If I upload myself into a computer and destroy my old body then the new "me" inside the computer will

Interestingly, we'll never have any evidence either way. If I upload myself into a computer and destroy my old body then the new "me" inside the computer will no doubt say "Oh yes, my experience was fully continuous", because it will have been from his. But that won't mean that my experience was fully continuous. And if the copy of me is not me then you won't be able to ask me because I'll be dead. To me the idea that my consciousness (as in, what I am currently experiencing) would get transferred across to a copy of me if I made such a copy and deleted the original seems to me rather implausible, though, especially as deleting the original seems to be optional in such a procedure. And no, I don't think quantum mechanics saves us.