How important is the local computer when using Claude Code for coding?

Just wondering how much of the work is done on the local computer. Am I able to speed Claude's work at all by buying a more powerful computer? My feeling is that is shouldn't matter much, as long as it's a decent computer and that the internet connection might matter more, but I hear some people talk about prompting and waiting for 15-30 minutes, so I wonder if it actually has an impact.