If one begins with non-relativistic Bohmian mechanics one notices that distant correlations in that theory are explained directly by the dynamics because the dynamics governs the total configuration of particle positions by a global law rather than governing the positions of individual particles by a local law. That means that where a particular particle goes may depend on where an arbitrarily distant particle is and, most strikingly, on what is being done to the distant particle. In the non-relativistic theory, determining where the distant particle is requires the use of absolute simultaneity: we mean where the distant particle is at the very same moment for which the velocity of the local particle is to be determined.
This feature of the theory is best illuminated by an example discussed by David Albert (1992: 155-60). In Bohmian mechanics, when one does a spin measurement on a particle that is not in an eigenstate of spin (so quantum mechanics makes only probabilistic predictions), the outcome of the measurement will depend, first, on exactly how the spin-measuring device is constructed and, second on the exact initial location of the particle.
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To be concrete, suppose we prepare a beam of particles in the state y-spin up, and then subject the particles in the beam to an x-spin measurement. Quantum mechanics predicts that half of the particles in the beam will exit the device going up and half going down. Bohmian mechanics further implies, for an apparatus like a Stern-Gerlach device, that the particles that exit going up were all initially located in the upper region of their wave function, and the half that go down were originally located in the lower half.
But the outcome of an x-spin measurement cannot always be determined in such a straightforward way, from the initial location of the particle being measured. As Albert points out, the situation becomes much more interesting if we make x-spin measurements on pairs of particles that are entangled. If, for example, we create a pair of electrons in a singlet state and then measure the x-spin of both, quantum mechanics predicts that the results will always be anti-correlated: one electron will exhibit x-spin up and the other x-spin down. But whether a particular electron exhibits a particular result cannot be determined simply by the initial location of that particle [by which I think Maudlin means the initial location of the particle in Bohmian mechanics, where particles have well-defined positions at all times]: if it could, then there would be a completely local account of the spin measurements, and they could not violate Bell's inequality (which they do). Suppose, for example, each of the pair of electrons is initially in the upper spatial region of the wave function [I think this assumption about the hidden Bohmian position may be crucial in hsi further discussion]. Then it cannot be that each electron will exit the device headed toward the ceiling if it enters the device in the upper region: that would violate the perfect anti-correlation. So what determines which electron will go up and which go down?
As Albert shows, the exact outcome of the experiment depends on which electron goes through the device first. If the right-hand electron is measured first, it will be found to have x-spin up and the left-hand electron x-spin down, but if the left-hand electron is measured first, one will get the opposite outcome. [But it may be that this is only true if you initially assume that 'each of the pair of electrons is initially in the upper spatial region of the wave function' as he said above; perhaps if each was initially in the lower region, the opposite would be true. And since the spatial location is part of the Bohmian hidden variables, we never actually know what region the electrons are in on a given trial, so this might not give us an experimentally testable way of determining whether the right electron had already been measured based on the direction of the left electron. The other possibility which occurs to me is that Bohmian mechanics does predict an experimental test of which electron was measured first, and that it matches nonrelativistic QM in this respect (since Bohmian mechanics is designed to always match nonrelativistic QM's predictions about experimental results), but that relativistic QM would predict something different.] And this holds no matter how far apart the two electrons are, and it holds without the action of any intermediary particles or fields traveling between the two sides of the experiment. So the behavior of the right-hand electron at some moment depends on what has happened (arbitrarily far away) to the left-hand electron. The dynamical non-locality of Bohm's theory is thereby manifest.
Since the exact outcome of the experiment depends on which x-spin measurement is made first, the notion of "first" and "second" has an ineliminable physical role in Bohm's theory. [I would think if he was concluding it had an 'ineliminable physical role' in orthodox QM, he would say so.] In the non-relativistic theory, which measurement comes first and which second is determined by absolute simultaneity. And if one is to transfer the Bohmian dynamics to a spacetime with a Lorentzian structure, one needs there to be something fit to play the same dynamical role. Since no such structure is determined by the Lorentzian metric, the simplest thing to do is to add the required structure: to add a foliation relative to which the relevant sense of "first" and "second" (or "before" and "after") is defined. The foliation would then be invoked in the statement of the fundamental dynamical law governing the particles.