This earlier paper might also help.
The big picture is that we are constructing something like special relativity's notion of time, for any partially ordered set. For any p and q in the set, either p comes before q, p comes after q, or p and q have no order relation. That last possibility is the analogue of spacelike separation. If you then have p, q, r, s..., all of which are pairwise spacelike, you have something resembling a spacelike slice. But you want it to be maximal for the analogy to be complete. The bit about intersecting all inextendible paths is one form of maximality - it's saying that every maximal timelike path intersects your spacelike slice.
Then, having learnt to think of a poset as a little space-time, you then associate Hilbert spaces with the elements of the poset, and something like a local Schrodinger evolution with each succession step. But the steps can be one-to-many or many-to-one, which is why they use the broader notion of "completely positive mapping" rather than unitary mapping. You can also define the total Hilbert space on a spacelike slice by taking the tensor product of the little Hilbert spaces, and the evolution from one spacelike slice to a later one by algebraically composing all the individual mappings. All in all, it's like quantum field theory constructed on a directed graph rather than on a continuous space.
my overall question is if the idea on its own naturally produce a single history, or does it still need some sort of "collapse" or other contrived mechanism to do so?
I find it hard to say how naturally it does so. The paper is motivated by the problem that the Wheeler-deWitt equation in quantum cosmology only applies to "globally hyperbolic" spacetimes. It's an exercise in developing a more general formalism. So it's not written in order to promote a particular quantum interpretation. It's written in the standard way - "observables" are what's real, quantum states are just guides to what the observables will do.
A given history will attach a quantum state to every node in the causal graph. Under the orthodox interpretation, the reality at each node does not consist of the associated quantum state vector, but rather local observables taking specific values. Just to be concrete, since this must sound very abstract, let's talk in terms of qubits. Suppose we have a QCH with a qubit state at every node. Orthodoxy says that these qubit "states" are not the actual states, the actuality everywhere is just 0 or 1. A many-worlds interpretation would have to say those maximal spacelike tensor products are the real states. But when we evolve that state to the next spacelike slice, it should usually become an unfactorizable superposition. This is in contradiction with the QCH philosophy of specifying a definite qubit state at each node. So it's as if there's a collapse assumption built in - only I don't think it's a necessary assumption. You should be able to talk about a reduced density matrix at each node instead, and still use the formalism.
For me the ontological significance of QCH is not that it inherently prefers a single-world interpretation, but just that it shows an alternative midway between many worlds and classical spacetime - a causal grid of quasi-local state vectors. But the QCH formalism is still a long way from actually giving us quantum gravity, which was the objective. So it has to be considered unproven work in progress.
It used to puzzle me that Scott Aaronson still hasn't come to terms with the obvious absurdity of attempts to make quantum mechanics yield a single world.
I should have realized what was going on when I read Scott's blog post "The bullet-swallowers" in which Scott compares many-worlds to libertarianism. But light didn't dawn until my recent diavlog with Scott, where, at 50 minutes and 20 seconds, Scott says:
"What you've forced me to realize, Eliezer, and I thank you for this: What I'm uncomfortable with is not the many-worlds interpretation itself, it's the air of satisfaction that often comes with it."
-- Scott Aaronson, 50:20 in our Bloggingheads dialogue.
It doesn't show on my face (I need to learn to reveal my expressions more, people complain that I'm eerily motionless during these diavlogs) but at this point I'm thinking, Didn't Scott just outright concede the argument? (He didn't; I checked.) I mean, to me this sounds an awful lot like:
Sure, many-worlds is the simplest explanation that fits the facts, but I don't like the people who believe it.
And I strongly suspect that a lot of people out there who would refuse to identify themselves as "atheists" would say almost exactly the same thing:
What I'm uncomfortable with isn't the idea of a god-free physical universe, it's the air of satisfaction that atheists give off.
If you're a regular reader of Robin Hanson, you might essay a Hansonian explanation as follows:
Although the actual state of evidence favors many-worlds (atheism), I don't want to affiliate with other people who say so. They act all brash, arrogant, and offensive, and tend to believe and advocate other odd ideas like libertarianism. If I believed in many-worlds (atheism), that would make me part of this low-prestige group.
Or in simpler psychology:
I don't feel like I belong with the group that believes in many-worlds (atheism).
I think this might form a very general sort of status catch-22 for contrarian ideas.
When a correct contrarian idea comes along, it will have appealing qualities like simplicity and favorable evidence (in the case of factual beliefs) or high expected utility (in the case of policy proposals). When an appealing contrarian idea comes along, it will be initially supported by its appealing qualities, and opposed by the fact that it seems strange and unusual, or any other counterintuitive aspects it may have.
So initially, the group of people who are most likely to support the contrarian idea, are the people who are - among other things - most likely to break with their herd in support of an idea that seems true or right.
These first supporters are likely to be the sort of people who - rather than being careful to speak of the new idea in the cautious tones prudent to supplicate the many high-status insiders who believe otherwise - just go around talking as if the idea had a very high probability, merely because it seems to them like the simplest explanation that fits the facts. "Arrogant", "brash", and "condescending" are some of the terms used to describe people with this poor political sense.
The people first to speak out in support of the new idea will be those less sensitive to conformity; those with more confidence in their sense of truth or rightness; those less afraid of speaking out.
And to the extent these are general character traits, such first supporters are also more likely to advocate other contrarian beliefs, like libertarianism or strident atheism or cryonics.
And once that happens, the only people who'll be willing to believe the idea will be those willing to tar themselves by affiliating with a group of arrogant nonconformists - on top of everything else!
tl;dr: When a counterintuitive new idea comes along, the first people to support it will be contrarians, and so the new idea will become associated with contrarian traits and beliefs, and people will become even more reluctant to believe it because that would affiliate them with low-prestige people/traits/beliefs.
A further remark on "airs of satisfaction": Talk about how we don't understand the Born Probabilities and there are still open questions in QM, and hence we can't accept the no-worldeaters interpretation, sounds a good deal like the criticism given to atheists who go around advocating the no-God interpretation. "But there's so much we don't know about the universe! Why are you so self-satisfied with your disbelief in God?" There's plenty we don't understand about the universe, but that doesn't mean that future discoveries are likely to reveal Jehovah any more than they're likely to reveal a collapse postulate.
Furthermore, atheists are more likely than priests to hear "But we don't know everything about the universe" or "What's with this air of satisfaction?" Similarly, it looks to me like you can get away with speaking out strongly in favor of collapse postulates and against many-worlds, and the same people won't call you on an "air of satisfaction" or say "but what about the open questions in quantum mechanics?"
This is why I think that what we have here is just a sense of someone being too confident in an unusual belief given their assigned social status, rather than a genuine sense that we can't be too confident in any statement whatever. The instinctive status hierarchy treats factual beliefs in pretty much the same way as policy proposals. Just as you need to be extremely high-status to go off and say on your own that the tribe should do something unusual, there's a similar dissonance from a low-status person saying on their own to believe something unusual, without careful compromises with other factions. It shows the one has no sense of their appropriate status in the hierarchy, and isn't sensitive to other factions' interests.
The pure, uncompromising rejection merited by hypotheses like Jehovah or collapse postulates, socially appears as a refusal to make compromises with the majority, or a lack of sufficient humility when contradicting high-prestige people. (Also priests have higher social status to start with; it's understood that their place is to say and advocate these various things; and priests are better at faking humility while going on doing whatever they were going to do anyway.) The Copenhagen interpretation of QM - however ridiculous - is recognized as a conventional non-strange belief, so no one's going to call you insufficiently humble for advocating it. That would mark them as the contrarians.