high court vs supreme court

high court and supreme court both are nouns.

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

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

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

Other synonyms of high court include state supreme court.

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