big league vs conference

big league and conference both are nouns.

Word	Noun	Adjective	Verb	Adverb
big league	Yes	No	No	No
conference	Yes	No	No	No

As nouns, conference is a hypernym of big league; that is, conference is a word with a broader meaning than big league:

big league: the most important league in any sport (especially baseball)
conference: an association of sports teams that organizes matches for its members

Other hypernyms of big league include league.

big league (noun)	conference (noun)
the most important league in any sport (especially baseball)	a discussion among participants who have an agreed (serious) topic
	a prearranged meeting for consultation or exchange of information or discussion (especially one with a formal agenda)
	an association of sports teams that organizes matches for its members