If you are cut to part time, after being full time, and especially if this makes your job MUCH LESS desirable, as in that say, with this drop in hours, you lose your company provided insurance, and you are really willing to wait a long time, at least month or two without any paycheck at all, and then MAYBE be approved for unemployment insurance, and you would rather draw the amount of weekly unemployment insurance for you in California, (check it out, it's still not a lot, even in CA) for six months or less, no more, no matter whether you've found a job by then or not.....
If you are on board with all that, then quit the job immediately. DO NOT work a minute at the part time hours. It's a big decision, and you have to make it fast. If you do this, you MAY be approved for unemployment insurance. It's not a sure thing, but this is your only possible move if you even sort of want to qualify.
If you work, even an hour, or a day or a week or two at the reduced rate, then you quit because after all they have cut your hours, you find you cannot meet your expenses on this reduced rate, etc. you will almost certainly NOT be approved for unemployment. Working at a job means you have accepted the terms of the job, the hours, the benefits, etc. The system does not give a hoot if your job is a small number of hours, any of that equivalency stuff, or if you are making less than you used to IF you have accepted the part time hours by working them, at all.
If you file for benefits, having put any work time between the hour cut and your quitting the job, you don't have much of any chance of approval. They told you how many hours you were going to get, and you took it. The system doesn't care that you took it because couldn't afford to be without a paycheck, or that you needed the money or any issue like that. You work it, you accepted it.
If you decide you don't like the uncertainty, that you think it would be safer to continue working at this job until you can find another one to go to, which incidentally I think is the wisest course for people to do in most cases, then I'd advise you WITHOUT QUITTING THE JOB to file a claim for unemployment benefits right away, explaining that your hours have been cut down. This will set up a claim. You will be able to see exactly what the weekly benefit you would draw if qualified will be. If the amount you are making working part time for your employer is less in a Saturday through Sunday week, (gross wages) than your weekly benefit amount, you will be eligible for partial unemployment insurance benefits while still working. All you say is that you are working all the hours your employer has for you, and they'll verify this. You have not quit the job. You'll work all you are allowed to work and draw some small amount of unemployment along with it, NOT, of course, your whole weekly benefit amount.
If you stick it out, this may cause your employer to do one of several things. Firstly, if they really want to get rid of you, they may trump up a reason and fire you. In these circumstances, with them being the driver of the action of you leaving, and they do it without a valid misconduct reason (performance is probably not going to be considered misconduct) your chances of being approved for unemployment benefits are pretty good. Much more than if you voluntarily quit the job, where the burden of proof is on you to show that you had a valid work related reason to quit the job.
They may just not fight any more if you continue to work there in spite of the hour cut, may give you a lay off due to lack of work or something and let you go and draw benefits easily until you find another job.
In either case, forget reading all the whys and wherefores of the unemployment laws and trying to guess whether or not you could draw benefits. Just remember, if you're going to quit, do it immediately, as soon as you've heard what they're going to do, in regard to your hours, before you work at the reduced number of hours at all.
If you have already worked any part of a reduced hours week, then my advice to you is DO NOT quit the job, keep working there until you find something else or they fire you. File that claim for unemployment right away, without quitting.
If you haven't quit yet, think about it right now. As I said, regardless of what they've done to you, how unfair it was, or all your faithful service, they have a right to reduce a person's hours, will always say it was in the best interests of the company. There's no labor law that prohibits them doing this. So unemployment insurance is your only recourse except finding a new job and moving on. And as I said, being approved after a quit is by no means guaranteed, it is a long process in which you receive no money for a long time, and then none at all unless approved. And emphasizing all I have said about this issue, if you've already worked the p.t. hours, do not quit.