quarrel vs debate

quarrel and debate both are nouns.

Word	Noun	Adjective	Verb	Adverb
quarrel	Yes	No	Yes	No
debate	Yes	No	Yes	No

As verbs, debate is a hypernym of quarrel; that is, debate is a word with a broader meaning than quarrel:

Other hypernyms of quarrel include argue, contend, fence.

quarrel (noun)	debate (noun)
an arrow that is shot from a crossbow; has a head with four edges	the formal presentation of a stated proposition and the opposition to it (usually followed by a vote)
an angry dispute	a discussion in which reasons are advanced for and against some proposition or proposal

quarrel (verb)	debate (verb)
have a disagreement over something	argue with one another
	discuss the pros and cons of an issue
	have an argument about something
	think about carefully; weigh