Comment by nl

Sorry for the delayed reply - it's taking some time for people to digest these!

Yes there are some extremely novel approaches. Scott Aaronson highlights[1] some of ones he's interested in, and I'd highlight:

> Matrix multiplication in O(n9/4) time—a rational exponent for once (!), and via a completely different approach than was used for O(n2.373) and so forth

And in his wife's comments on USG (which she worked on):

> The UGC proof invents a completely new bizarre code with a noise test. It’s some crazy recursive construction. It’s not the long code, not the short code – some alien craziness

[1] https://scottaaronson.blog/?p=10169