1. They can be modified by degree adverbs like "very" (A very entertaining show), but you can't say *A very living composer.
2. They can occur as complement to complex-intransitive verbs like "become", (It became quite entertaining) but you can't say *The language became quite living.
3. They can occur as complement to complex-transitive verbs like "find" (I found it quite entertaining), but you can’t say *I found it quite living.
Other examples of verbs pre-modifying nouns are:
A sleeping child. An approaching train. Some melting marshmallows.
Which is why some present participles like “living", "sleeping", "approaching" and "melting" are best categorised as VPs when they are pre-modifying nouns.