I don't quite understand how this works. I get that the League of Entropy is a verifiable timestamped network source of randomness, but how does this let you set a future time for decrypting something that you encrypt now. How do you depend on a quality of the random source in the future, when by definition that can't be predictable?
It's not using the future random numbers to decrypt the value. It's making use of the underlying protocol that the League of Entropy is using primarily but not exclusively for calculating fair random numbers.
I believe (admittedly from skimming and educated guesses) the setup is that the League of Entropy network is a set of computers operated by different people which each have published public keys, and the participants regularly do multi-party computation to create random numbers that provably can not be manipulated by any non-majority subset of the group. Besides calculating verifiably random numbers, the computers will also decrypt content for anyone if you send it content encrypted to their public keys and the encrypted content contains a time range that includes the present. (Maybe their main protocol for random numbers requires this protocol to be implemented?) Then you can do time-lock encryption by encrypting some content, splitting the key using a threshold scheme, then separately encrypt each part of the key to a different participant of the League of Entropy network bundled with a requested time-range for decryption to be allowed. It works as long as enough League of Entropy network participants stay online and enough of them don't defect or get hacked.
Yeah, it's not exactly how it works, but close. The League's node ran together a distributed key generation that created a secret key for the whole League so that it's sufficient to have only a threshold number of nodes to recreate that secret key. But since it's using threshold cryptography, it doesn't even ever need to recreate the secret, it can simply aggregate "partial signatures" into valid signatures for the group's public key. The timelock scheme itself relies on so-called "identity-based encryption", so that you can encrypt a plaintext towards the signature of a specific message. And you know in advance that the League is going to sign round number X at time Y, so you can encrypt towards the signature of the round number X, and once it's produced, that signature allows to decrypt the ciphertext. Pairing-based cryptography is kinda magic on that front :)
