Imagine a team assembling a jigsaw puzzle from random pieces: how many simultaneous moves are needed to complete the picture? If the pieces are independent, they all go in at once. If each piece depends on the previous one, you have to proceed one at a time. The same goes for systems: some states come together instantly, while others demand a long, sequential effort.
The measure proposed by the author is called parallel depth. It is the minimum number of simultaneous steps on a multi-processor computer needed to reproduce a typical state of the system. For a memoryless coin toss, the depth is constant: all bits are independent and generated at once. For a coin with memory, the depth is logarithmic: the product of independent factors can be computed using a divide-and-conquer approach.
Diffusion-limited aggregation (cluster growth via random walks) is a different story. Here, each particle must wait for the previous one, and depth grows linearly with the number of particles. Generating such a cluster is P-complete: no fast parallel algorithm exists. Previously, it was unclear how to distinguish systems that merely store complex correlations from those that actually perform long computations. Now it is clear: depth is small for processes with memory, but large for sequential processes like diffusion-limited aggregation. This measure can be applied to classify complex systems in physics, biology, economics, anywhere states are described by probability distributions.
If an alien researcher measured the complexity of the Solar System by entropy, they would focus on the Sun, where almost all the mass and entropy are concentrated. But a depth measurement would point to the surface of a small planet, Earth: the biosphere, whose complexity arises from a long evolutionary history, not from volume.