A team assembling a jigsaw puzzle from random pieces: how many simultaneous moves does it take to complete the picture? If the pieces are independent, they all go down at once. If each piece depends on the previous one, you have to work one at a time. So it is with systems: some states assemble instantly, others require a long sequential effort.

The measure proposed by the author is called parallel depth. It is the minimum number of simultaneous steps on an idealized computer with many processors needed to reproduce a typical state of the system. Depth does not ask how much total computation or randomness is involved, only how much parallel time. For a memoryless coin toss, depth is constant: all bits are independent and generated simultaneously. For a coin with memory, depth is logarithmic: the product of independent factors can be computed using a divide-and-conquer approach.

Diffusion-limited aggregation (cluster growth by random walk) 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. Depth is maximal: if a large system contains one small but very complex subsystem, the depth of the whole system is determined by that subsystem, not by averaging over volume.

Previously, it was unclear how to distinguish systems that merely store complex correlations from systems 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. The measure can be applied to classify complex systems in physics, biology, economics, anywhere states are described by probability distributions. The main issue is that depth cannot be computed exactly; it can only be bounded from above by demonstrating increasingly efficient algorithms. But already it helps reveal where genuine computation is hidden within a system.