Q: What is the age that kids can be stay at home alone ?

Here are some information from https://www.childwelfare.gov/pubpdfs/homealone.pdf with some highlight for your convenience:

Some parents look to the law for help in deciding when it is appropriate to leave a child home alone. Many States’ child protection laws classify “failing to provide adequate supervision of a child” as child neglect, but most of these States do not provide any detail on what is considered “adequate supervision.” In some States, it is considered neglect when a child has been left without supervision at an inappropriate age or in inappropriate circumstances, after considering factors such as the child’s age, mental ability, physical condition, the length of the parent’s absence, or the home environment—any combination of factors that creates a situation that puts the child at risk of harm. Only three States currently have laws regarding a minimum age for leaving a child home alone. Illinois law requires children to be 14 years old before being left alone; in Maryland, the minimum age is 8, while in Oregon, children must be 10 before being left home alone. Many States offer guidelines for parents that can assist them in determining when it’s appropriate for them to leave their children home alone. (See the end of this factsheet for a listing of States that have posted guidelines on the Internet.) For information on laws and guidelines in your State, contact your local CPS agency. If you need help contacting your local CPS agency, call Childhelp at 800.422.4453.

You can find some guideline/resources from state's website:
Arizona: https://www.azdes.gov/landing.aspx?id=9694
California: http://www.cde.ca.gov/sp/cd/re/caqhomealone.asp
Colorado: http://www.colorado.gov/cs/Satellite/CDHS-ChildYouthFam/CBON/1251617279471
Connecticut: http://www.ct.gov/dcf/lib/dcf/child_welfare_services/pdf/leaving_your_ch...
Delaware: http://kids.delaware.gov/faqs.shtml
District of Columbia: http://dc.gov/DC/CFSA/About+CFSA/Who+We+Are/FAQs#1
Florida: http://www.dcf.state.fl.us/programs/childwelfare/caregivers/ whentoleaveyourkidshomealone.shtml
Georgia: http://dfcs.dhs.georgia.gov/safety-precautions-when-leaving-children-hom...
Idaho: http://www.211.idaho.gov/elibrary/HomeAlone.html
Illinois: http://www.state.il.us/dcfs/docs/leave.pdf
Indiana: http://www.in.gov/dcs/files/Home_Alone_Brochure.pdf
Kansas: http://www.dcf.ks.gov/services/PPS/Documents/PPM_Forms/Appendices/Append...
Maine: http://www.mainekids-kin.org/Tip%20Sheets/9%20-%20Is%20your%20child%20re... to%20stay%20alone.pdf
Maryland (Anne Arundel county): http://www.aacounty.org/dss/children.cfm
Massachusetts: http://www.lawlib.state.ma.us/subject/about/childabusefaq.html#home
Missouri: http://extension.missouri.edu/extensioninfonet/article.asp?id=2250
Montana: http://www.dphhs.mt.gov/publications/isyourchildreadytobehomealone.pdf
Nebraska: http://dhhs.ne.gov/children_family_services/Pages/children_family_servic...
New Hampshire: http://www.dhhs.nh.gov/dphs/documents/home-alone.pdf
New York: http://ocfs.ny.gov/main/prevention/faqs.asp#supervision
North Carolina: http://www.ncdhhs.gov/contacts/faqs.htm
North Dakota: http://www.pcand.org/images/stories/Home_Alone_Brochure_lowres.pdf
Ohio: http://jfs.ohio.gov/OCTF/leaving_your_child_home_alone.pdf
Oklahoma: http://www.okdhs.org/NR/rdonlyres/83731D8C-EA74-4CA4-840F-498451D77245/0... GuidelinesforChildrenLeftAlone_aging_2004.pdf
Pennsylvania: http://www.phila.gov/dhs/PDFs/homeAlone.pdf
Texas: https://www.dfps.state.tx.us/Child_Protection/About_Child_Protective_Ser... faqchildalone.asp
Virginia: http://www.dss.virginia.gov/files/division/dfs/cps/intro_page/publicatio...
Washington: http://www.dshs.wa.gov/pdf/media/052912HomeAlone.pdf