It would be pretty trivial to paraphrase the output wouldn't it?

Change one character and the hash will not match anymore...

There are the issues others mentioned, but also you could write something word for word of what an LLM says.

It’s statistically unlikely, but possible

Hashes of every answer to every question and every variation of that question? If that were possible, you’d still need to account for the extreme likelihood of the LLM providing a differently worded answer (it virtually always will). This isn’t how LLMs or hashing algorithms work. I think the answer is that teachers need to adjust to the changing technological landscape. It’s long overdue, and LLMs have almost ruined homework.

Ever heard of the meme:

"can I copy your homework?"

"yeah just change it up a bit so it doesn't look obvious you copied"

Local models are possible and nothing in that area of development will ever publish a hash of their output. The huge frontier models are not reasonably self-hosted but for normal K-12 tasking a model that runs on a decent gaming computer is sufficient to make a teacher's job harder. Hell, a small model running on a newer phone from the last couple of years could provide pretty decent essay help.

There’s an actual approach where you have the LLM generate patterns of slightly less likely words and then can detect it easily from years ago. They don’t want to do any of that stuff because cheating students are their users.

I see the problem you face, but I don't think it's that easy. It seems you can rely on hashes being noisy and alter questions or answers a little bit to get around the LLM homework naughty list.

Just ctrl-f for an em dash and call it a day.