Comments on entropy and the direction of time in Landau and Lifshitz's Statistical Mechanics In Landau and Lifshitz's Stat Mech Volume I is the comment:

However, despite this symmetry, quantum mechanics does in fact involve an 
  important non-equivalence of the two directions of time. This appears in connection with the interaction of a quantum object with a system which with 
  sufficient accuracy obeys the laws of classical mechanics, a process of fundamental significance in quantum mechanics. If two interactions A and B with 
  a given quantum object occur in succession, then the statement that the probability of any particular result of process B is determined by the result of 
  process A can be valid only if process A occurred earlier than process B? 
Thus in quantum mechanics there is a physical non-equivalence of the two directions of time, and theoretically the law of increase of entropy might be its macroscopic expression.  In that case, there must exist an inequality involving the quantum constant $\hbar$ which ensures the validity of this law and is satisfied in the real world.  Up to the present, however, no such relation has been at all convincingly shown to exist.

Has there been such convincing work relating the direction of time and $\hbar$ since these comments were first made (somewhere between 1937 and 1975 by the dates in the prefaces)?
 A: Peeter, there is a paper on the dissipation in quantum theory by Callen and Welton from 50ies:

Irreversibility and Generalized Noise. H. B. Callen and T. A. Welton. Phys. Rev. 83 no. 1, pp 34-40. Caltech e-print.

where they derive the rate of power dissipation (irreversible) from other ideas of quantum theory. It is not rigorous though - they use the "Fermi golden rule" which itself is not time reversible, so the question then becomes how does the irreversible golden rule follow from Schrödinger's equation, which is reversible.
A: Landau's argument is basically a swindle for two reasons. The first reason is that he's trying to present this without talking about measurement or wavefunction collapse, which would make it sound like a fundamental statement about physical laws rather than a statement about one possible philosophical interpretation of quantum mechanics.
To see that this is misleading, note the use of the word "probability." Probability and randomness are tricky to define. One of the standard approaches is to talk about a sample space. For example, WP has:

In probability theory, the sample space of an experiment or random trial is the set of all possible outcomes or results of that experiment.

Note those words like "experiment," "trial," and "outcome." To give meaning to the word "probability," we would typically do something like the following: prepare an ensemble of systems in the same state, do an experiment in which we measure some observable, and count how many times we get different results. But this is clearly a measurement process. The most common definitions of the words Landau is using occur in the context of measurement, and if he had in mind some definition that didn't require a notion of measurement, the burden would be on him to spell out what those definitions were.
In fact, Landau's statement about events A and B is really a more general statement about what happens when we make measurements or observations on any random process, not just a process that's random because of quantum mechanics. Let's take a classical system like a flipping coin. The coin is described to essentially perfect precision by Newton's laws, but the outcome is random because of its extreme sensitivity to initial conditions. Let A be the event that when the coin comes to rest on the tabletop after the flip, it's heads-up. Let B be the event that, one second after that, it's heads-up. Landau's argument applies just as well to this story as to his story of a quantum-mechanical measurement; all we have to do is replace "quantum object" with "coin" and "a system which with sufficient accuracy obeys the laws of classical mechanics" with "a system whose behavior is predictable because it is not too sensitive to initial conditions."
Landau says:

In that case, there must exist an inequality involving the quantum constant $\hbar$ which ensures the validity of this law and is satisfied in the real world.  Up to the present, however, no such relation has been at all convincingly shown to exist.

Here Landau is simply demonstrating that his argument isn't quantum-mechanical.
Now we come to the second element of the swindle: the words "is determined by." These have not been defined. In everyday life, a person asked to describe the coin flip would say that B "is determined by" A, since A was where the coin "decided how to lie," and B just describes the coin "staying where it was." But of course we can time-reverse Newton's laws, in which case B caused A. There's a dash of mystery here, but it's a classical mystery, not a quantum-mechanical one: it would be consistent with the laws of physics for the coin to jump off of the table, rise into the air flipping end over end, and land on your thumb -- and yet we never observe such processes in nature.
In summary, there is nothing quantum-mechanical about Landau's argument, and it really just boils down to an idea about measurement in general, which is that measurement is not time-reversible. One way of understanding why measurement is not time-reversible is that measurement involves the processing and recording of information, e.g., by a human brain. Thermodynamically, a human brain wouldn't be able to exist if the universe were in a state of maximum entropy. For reasons unknown to us, the big bang in our universe was a low-entropy big bang, and therefore the universe is not yet in a state of thermal equilibrium. This is the origin of the psychological arrow of time and the notion that measurement isn't time-reversible.
A: Each process of scattering is accompanied with infinite number of soft photons, i.e., it is generally an inelastic process. It is extremely difficult (impossible) to reverse such a process. Even in a thermal bath (i.e. with presence of soft photons in the initial state) it is impossible to reverse exactly scattering, so real processes are irreversible due to being inelastic.
Unfortunately, people reason in terms of elastic processes, even in QED where it is proven that no elastic processes are possible. This is the root of not only "time direction issue", but also of UV and IR problems in calculations.
A: The key here is that quantum mechanics is incomplete and cannot describe the measurement process still less 'the arrow of time' associated to measurements.
The Brussels School has developed an extension of quantum mechanics that addresses such questions. Next you can find an introduction
http://www.ph.utexas.edu/~gonzalo/3bgraphs.html
The 'microscopic' version of the second law is given as $\Im(\Theta)\leq 0$ where $\Theta$ is the complex extension of the unitary evolutor $U$ for dissipative systems.
$\hbar$ does not play any special role in the theory, because irreversibility is also present at the classical level, i.e. when $\hbar \to 0$. The fundamental constant here is $k_\mathrm{B}$.
