Cosma links to this article about a 'quantum solution to the arrow-of-time dilemma' and suggests to read it carefully.
I think I read it already a year ago as this preprint and it has several interesting ideas; In particular I liked the passages about Borel's argument.
But I was also thinking about the hidden assumption of this paper.
As far as I understand it, the author (implicitly) assumes that memories are about the past and in a paper about the 'arrow-of-time dilemma' this should be explained not assumed.
In other words, the author suggests that we could live in a time-symmetric world and we just do not have memories about "phenomena where the entropy decreases"; This is all fine, but in my opinion one needs to explain then why we only have memories about the past but not the future, without "an ad hoc assumption about low entropy initial states".
E.g. in the conclusions we read that "we could define the past as that of which we
have memories of, and the future as that of which we do not have any memories" (*), but the author does not ask or answer the question how there can even be such a thing as the future, of which we have no memories, in a time-symmetric world.
update: Sean wrote about the paper also and in the comments the author responds.
I am glad that at least one comment agrees with my own reading; And it seems that Nick Huggett has some interesting papers I should read.
update2: The author responds to Huw Price (who raised the same objection I did).
update3: Last, but not least, Dieter Zeh comments on the paper as well. I wrote about his book about 'the direction of time' previously.
PS: The question "if the world would run 'backwards', would we even notice it?" was raised and discussed previously on this blog.
(*) As it stands this statement is of course contrary to the usual convention(s). From what we know about physics, most of those events of which we have no memory are space-like to us and not in our future. And of course there are even events in our past of which we have no memories. E.g. we know for sure that Philip Augustus of France spent a night with Ingeborg of Denmark, but we have no documents or memories about the mysterious events of that night.