A caveat here. They may not even give you the time of day if you call and ask a general question like this. Look on your state's websites, and look in particular at the "general questions and answers" section. But in most cases, in most states, in fact, in all states, to qualify for unemployment benefits, you must be able, available and actively seeking other work. So probably no, even though your daughter is medically fragile, and this is documented, if you quit your job to care for her, there isn't much chance AT ALL that you will be able to draw unemployment benefits. As they continuously point out, unemployment insurance is NOT sick leave, medical insurance or for those who are not able to work. Have you considered FMLA to care for her, perhaps intermittent FMLA?