I don't know about other states, but California provides up to 6 weeks of Paid Family Leave. "Benefit amounts range from $50 to $1,173 per week and represent about 55 percent of your previous weekly earnings (maximum reimbursement amount is $1,173 per week)."

http://edd.ca.gov/Disability/About_PFL.htm

I wish it were longer, and I wish it paid more, and I wish it were federal, but it's better than nothing.