DD is only 3 so take it with a grain of salt
In our state (FL) there is no set age - it is up to the parents' discretion.
I would say it depends on your dc. I dunno as I would let an 8 year old babysit younger siblings (too much responsibility for being so young imo) but I would totally allow a mature 8 year old to stay home for very short periods -- like a quick trip to the store, picking another child up from an activity... a pedicure
or something of that nature.
I would make sure dc knew, understood, and agreed with certain "rules" I would feel comfortable setting for a child so young -- don't answer the phone or tell anyone you are home alone, keep the door locked, no answering the door unless it is a family member (like grandmom or whatever), no going outside, no using appliances other than maybe the toaster (or something simple and pretty fire-proof .... things of that nature.
My best friend's sons were 7 and 9 when she left them alone for short periods (like quick trips to the corner store etc) and she is a police officer so she obviously felt this was okay.
I think it depends on the child's maturity level and well, only you and your child know that
My mom would leave my sister and I alone after school until she got home from work (from only about 3pm to 530pm) when we begggggggggggggggggggged her not to have a babysitter anymore. We were about hmmm... 7 and 11? We did fine, but my sister was really mature as well and we had next door neighbors that were friendly and helpful in case of emergency (of which there never was).
Good luck in your decision.