We would have broken some fundamental laws, because to multiply two n x n matrices, it takes n^2 time to write down the answer. So the better question to ask i

We would have broken some fundamental laws, because to multiply two n x n matrices, it takes n^2 time to write down the answer. So the better question to ask is: What happens if we find a practical algorithm (and a theoretical approach) that lets us do matmul in only n^2 time? And the answer is - well, some important things get faster and we're able to solve larger problems in things like optimization, simulation, deep learning, etc., or save a lot of time and money doing them. We'd go from about n^2.8 to n^2, which, let's say for a 1M x 1M matrix, is about 64k times faster. That's really nice. But the speedup for more common sized matrices is smaller - 256x faster for a 1024x1024 matrix. It would be a very important thing that would cause us to re-examine the use of matrix multiplication as a primitive for more things, and would have very important practical implications - and at the same time, would also kind of look like 15 years of Moore's law, not that we're guaranteed to have 15 years of that. So, "big" but not necessarily "totally reshape the world".