Home schooling is very state specific, and can depend on the state and what type of program you are using,
In TN for example, if you do not use a regular program, but pick your own books and don't use a program grading, you do have to have a college degree when you get into high school program, but if you use a set program you only have to have a high school dipolma.
Also if you use or do it under the umbrella of a home school program that is acredited, there is no notification of the state required at all.
But that various from state to state and changes yearly on the laws.
For example we use Seton for our home school, it is a Catholic School, they even have a school number when the kids graduate to use when applying to college. So as such in our state, you are just in the program, and no notification is required.
This web site, ( home legal defense fund) which is a must to belong to since local school systems often break the law trying to force you to do things not required by state law.
But this site has state by state laws as to what is required
Homeschool: HSLDA-Home School Legal Defense Association: