That's not really a problem. You don't hash the entire video, but do something resembling a merkle tree. i.e. look at a torrent, they're identified by a hash but you can download and verify a random chunk
Multiple levels of the tree can be stored throughout the video file. The first level after the root can be for major sections, like 5 minute segments. The next levels are then at the start of each 5 minute segment, giving hashes for one second chunks.
If the root hash checks out, we get the 5-min hashes. If they check out, we get the hashes for the first 5-min block, and if those hashes check out, we start to play the video, validating every second of it against a one second hash from the 5 min block. Then we get the next 5-min hash block and so on.
Kind of thing.