My desire for "real breakthroughs ... in chaos theory" to be exactly every 50 years (to the day! the minute!) is extremely high.

If you could comment. Is it true that "classic" quantum field theory can't describe chaos, because the operators are linear?

Yep. Quantum chaos, at least insofar as I was involved with it, is about semi classical analysis of quantum systems which would have chaos if they were classical.

Remember quantum mechanics started with the Bohr atom which was an approach that would generalize to any system that has quasi periodic (solar system in the short term) dynamics. Circa 1917 Einstein wrote a paper that said that this approach wouldn’t generalize because Poincaré proved that chaos is generic, soon after that we got Schrodinger’s equation and similar approaches. (Although that Einstein paper killed a line of development it barely got cited for 50 years)

Later on there were mathematical developments in quantum chaos such as Gutzwiller’s trace formula and also numerical work where people discovered various properties of the quantum levels of systems that were classically chaotic.

Linear operators in quantum mechanics don’t precluded observing chaos in the real world because the true “long term” is very long indeed. That is, classically or quantum mechanically if you wait some really absurdly long time (say much more than 10^120 years) the universe is expected to come arbitrarily close to the state it is in today. See

https://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theor...

Though note that the recurrence theorem applies to a finite phase space. An expanding universe such as ours is not subject to that. We just get heat death instead.