1. He made them be quiet in the library.

2. He made them quiet in the library.

3. He had them be quiet in the library.
4. He had them quiet in the library.

5. He watched them be quiet in the library.

6. He watched them quiet in the library.

(Which ones are correct in the sentences above?)

All are correct, but these would be more commonly used: 1, 3, and 5.

Out of the six sentences provided, the correct ones are:

1. He made them be quiet in the library.
3. He had them be quiet in the library.

In these sentences, the combination of the verb "made" or "had" with the structure "them be quiet" is appropriate. The use of the infinitive verb "be" after the object "them" indicates the action or state they were made or had to do.

The following sentences are not grammatically correct:

2. He made them quiet in the library.
4. He had them quiet in the library.

In these sentences, the adjective "quiet" is used without the verb "be." This usage is not standard grammar, as it omits the linking verb "be" that would clarify the existence of the state of being quiet.

Regarding the last two sentences:

5. He watched them be quiet in the library.
6. He watched them quiet in the library.

Both sentences are grammatically correct, but they convey slightly different meanings.

Sentence 5 emphasizes the act of observing the action of "them being quiet." In this case, the verb "watched" is complemented by the infinitive verb "be," indicating the action being observed.

Sentence 6 focuses on the state of being quiet, emphasizing the observation of their quietness in the library. Here, the verb "watched" is complemented by the adjective "quiet" itself.

In summary, the correct sentences are 1, 3, 5, and 6.