This is an important theorem which we sort of independently discovered with Daniel when describing boson correlations of classical states.[1]
It is also a subtle one that deserves profound attention. It posits that random-variable observations can be recognized as independent with correlations due to a latent parameter from a property of interchangeability. The way I understand it is that correlations of different types are causal and with markedly distinct characteristics, such as memory, decay in time (or space), etc., while those from de Finetti look like the type of "spurious" correlations we identified with Daniel but instead of being due to aggregates of heterogenous samples, they are due to lack of knowledge or the nature of one's prior.
There is a quantum version known as "Hudson-Moody" theorem,[2] which, according to C. Caves et al.,[3] «relies on advanced mathematics».