high court vs state supreme court

high court and state supreme court both are nouns.

Word	Noun	Adjective	Verb	Adverb
high court	Yes	No	No	No
state supreme court	Yes	No	No	No

As nouns, high court and state supreme court are synonyms defined as:

high court and state supreme court: the highest court in most states of the United States

Other synonyms of high court include supreme court.

high court (noun)	state supreme court (noun)
the highest court in most states of the United States	the highest court in most states of the United States