The complexity class $\mathrm{MIP^*}$ includes the set of languages that can be efficiently verified by a classical, polynomially-bounded verifier, engaging with two quantum provers that can share (potentially infinite) entanglement, but are otherwise forbidden to signal to each other. The complexity class $\mathrm{RE}$ includes recursively enumerable problems, the Halting Problem being a complete problem therein.
A recent breakthrough of Ji, Natarajan, Vidick, Wright, and Yuen in quantum complexity theory is $\mathrm{MIP^*=RE}$. This is, in a way, a quantum analog of the classical result that $\mathrm{MIP=NEXP}$, but shows that shared quantum entanglement in a potentially infinite-dimensional Hilbert space is much much more powerful than that allowed in the (unentangled) classical setting.
Rolling back the implications of Tsirelson and others, this implies a negative answer to Connes' Embedding Problem, which is a major problem in algebra theory.
I can get my head around the complexity class $\mathrm{MIP^*}$ and that of $\mathrm{RE}$. Thanks to Yuen's lecture at the IAS, I vaguely understand Tsirelson's problem in relation to CHSH games, etc., and how it could relate to $\mathrm{MIP^*}$. Also I was able to follow a lot of Slofstra's lecture, where he mentions that Tsirelson's problem is related to finding magic squares in $\mathbb{Z}_2$, and emphasizes the relation to solving a specific word problem for finitely presented groups. Famously such word problems can be algorithmically unsolvable.
However, I know very little of even the statement of Connes' embedding problem, and I don't know how to analogize the problem to something that I'm familiar with.
According to Wikipedia, "Connes' Embedding Problem asks whether every type II1 factor on a separable Hilbert space can be embedded into some $R^\omega$."
Is the CEP itself akin to an algorithmic problem as in the word problem for a finitely generated groups? Or is it much more subtle than that?
Basically I'm looking for a statement of Connes' embedding problem that is a bit more accessible than the Wikipedia article?