Unless they changed it, in my state every week you had to submit a list of all the places you applied at. This was quite some years ago when I had just gotten out of the military and hadn't found a new job yet, but I doubt it has changed significantly.
I would assume the reason they don't require you to have a resume hanging out online is the vast majority of people on unemployment aren't really expected to apply for the kind of jobs that require a resume. Getting a job at McDonalds doesn't exactly require one (or it shouldn't). Those few that are actually qualified for jobs not at the bottom of society either a) are not very good at their jobs or b) very rare.
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