living quarters vs housing

living quarters and housing both are nouns.

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

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

Other hypernyms of living quarters include living accommodations, lodging.

living quarters (noun)	housing (noun)
housing available for people to live in	structures collectively in which people are housed
	a protective cover designed to contain or support a mechanical component
	stable gear consisting of a decorated covering for a horse, especially (formerly) for a warhorse