dormitory vs living quarters

dormitory and living quarters both are nouns.

Word	Noun	Adjective	Verb	Adverb
dormitory	Yes	No	No	No
living quarters	Yes	No	No	No

As nouns, living quarters is a hypernym of dormitory; that is, living quarters is a word with a broader meaning than dormitory:

dormitory: a college or university building containing living quarters for students
living quarters: housing available for people to live in

Other hypernyms of dormitory include building, edifice, quarters.

dormitory (noun)	living quarters (noun)
a college or university building containing living quarters for students	housing available for people to live in
a large sleeping room containing several beds