An example of this is that I have seen both 来週テスト and 来週のテスト. Both of them
mean "next week's test".

My issue is that I have no clue when to use which or what makes them different
from each other. In what situations can you leave out the の?

Edit: Just to clarify, the example I've mentioned above is one of many
instances I've heard, as in when listening to shows to better grasp spoken
Japanese (opposed to textbook). I'm trying to get a general guideline as to
situations when one leaves it in, and when one takes it out.
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question
particles
Why do you leave out the の in some cases?
The meaning "来週テスト" is often used as a casual or abbreviated meaning among the
general Japanese population, often in the sense of "来週テストがある"The correct
grammar is "来週 **、** テストがある" or "来週 **は** テストがある"

来週のテスト means (a test that is next week). Or next week's test

This is very difficult for even native speakers to understand sometimes.
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It's determined by the will of the speaking human. There are no rules and no
guidelines. There is no need in specification, because it is used the
semantics of kanji to understand the sense of the phrase. It's passed in the
official documents, written speech, newspaper articles etc.
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Here is the critical difference:

  * **来週の** always modifies a **noun** (aka "adjectival").
  * **来週** always modifies a **verb** (aka "adverbial").

来週, 明日, 昨日 and so on do not take any particle when used adverbially. (In
English, words like "tomorrow" do not take prepositions when used as an
adverb, either. For example, you say "I'll do it tomorrow" rather than "I'll
do it on tomorrow".)

テスト is a noun, so you should always use 来週 **の** テスト to say "next week's
exam".

On the other hand, 来週テスト is simply an incorrect phrase at least in formal
Japanese. In colloquial Japanese, you can understand this as an abbreviated
sentence where がある is omitted (note that ある is a verb), i.e., it can be an
abbreviated sentence meaning "I have an exam next week". Either way, it never
means "next week's exam".

Note that Japanese is a language that strictly distinguishes noun-modifying
expressions and verb-modifying expressions. See also: [using の with と,で, から,
まで](https://japanese.stackexchange.com/q/33605/5010)
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