Is it possible to securely combine multiple hashes without hashing them? Let's say we have 1000 files and 1000 already computed cryptographic hashes for those files. Now we want to derive a single hash that authenticates all of them. Is there a more efficient way to do this than just hashing the concatenation of all those hashes (and thereby presumably calling the hash round function about 1000 times), without sacrificing security? What about also verifying the sequence?
What you're looking for is called a Merkle tree. BLAKE2b, a modern hash and an evolution of one of the SHA-3 finalists (BLAKE), supports tree hashing natively.

Edit: This may or may not actually be what you're looking for. Initially hashing the tree will take more work ($\mathcal{O}(n\log(n))$ operations) than just hashing the set of hashes, but subsequent partial updates will take less work ($\mathcal{O}(\log(n))$ operations). If you're ever updating the list, a Merkle tree is probably more efficient.