Step 1: Recall the definition of a noun.
A noun is a word that names a person, place, thing, or idea. We scan the sentence: "His involvement in music earned him good recognition across the State."
Step 2: Identify each noun.
"Involvement" (abstract noun), "music" (abstract noun), "recognition" (abstract noun), and "State" (proper noun) are all nouns in this sentence.
Step 3: Eliminate non-nouns.
"Earned" is a verb, and "across" is a preposition, ruling out options (1), (2), and (3). Option (4) lists all four correct nouns.
\[ \boxed{\text{involvement, music, recognition, State}} \]