Quickly walking to the store (phrase or clause why?)

Until next week (phrase or clause why?)

Yet they tried (phrase or clause why?)

A) phrase

B) phrase

C) clause

Explanation:

First we need to make a distinction between a phrase and a clause.

A phrase is a part of sentence, a set of words, that lacks in subject, verb or both so it can never stand on its own. It can have several functions in a sentence depending on the piece of information it carries.

A clause is a part of a sentence that has a subject and a verb and can either stand on its own as a complete sentence (independent clause) or not (dependent clause).

A) This is the phrase because it doesn't have a subject and "walking" isn't a verb, but a gerund. Most likely it's an adverbial phrase telling how or why something happened. For example:

Quickly walking to the store, Ian sprained his ankle.

B) This is also a phrase because it only gives us a time frame; we don't know who will (or won't) perform an action nor what will (or won't happen). For example:

I won't leave the house until next week.

C) This is a clause; it has subject (they) and verb (tried). However it doesn't make a complete sentence, so it's a dependent clause:

They were exhausted yet they tried.