PHYSICAL REVIEW X 8, 031013 (2018)
Jayne Thompson, Andrew J. P. Garner, John R. Mahoney, James P. Crutchfield, Vlatko Vedral, and Mile Gu
...Consider a cannonball in free fall. To model its future trajectory classically, we need only its current position and velocity. This remains true even when we view the process in reverse time. This exemplifies causal symmetry. There is no difference in the amount of information we must track for prediction versus retrodiction.
However, this is not as obvious for more complex processes. Take a glass shattering upon impact with the floor. In one temporal direction, the future distribution of shards depends only on the glass’s current position, velocity, and orientation. In the opposite direction, we may need to track relevant information regarding each glass shard to infer the glass’s prior trajectory.
 J. P. Crutchfield, Between Order, and Chaos , Nat. Phys. 8, 17 (2012).
This paper shows a quantum causal asymmetry that does not exist classically and uses a cannonball as an example of classical time reversal symmetry of prediction and retrodiction. However, including the atmospheric friction around the cannonball trajectory results in the same classical versus quantum complexity dilemma as this actual cannonball trajectories as a painting in1628 by Diego Ufano shows.
Even the simplest actual classical cannonball trajectory involves much more classical than quantum information since the trajectory is continuous and infinitely divisible but chaotic due to atmospheric friction. However, the quantum trajectory involves discrete jumps or hops and quantum therefore ultimately limits the information needed for retrodiction. However, the price to pay for that quantum limit is in a limited uncertainty while classically, there is no limit to the uncertainty and therefore the information is unbounded.
The cannonball trajectory makes up a causal set of precursors and outcomes and are all predicated on atmospheric eddies at a higher resolution. Eventually, a discrete quantum limits the information for quantum retrodiction and so provides a kind of quantum arrow of time. Although not discussed in this paper, it is quantum phase decay that brings quantum and classical retrodiction together as one.