Includes [sidechain] approaches, notary schemes, hash time hash-locks. E.g. state channels are off-chain side channels used to implement payment channels by offloading transactions. In a payment channel, participants interact, collect cryptographically signed messages to update the current state w/o publishing to main chain. Good for when multiple transactions are to be summed up into a single one.
Cosmos, Polkadot - Relay Chains
E.g. Hyperledger Cactus
Cactus allows a party or a set of parties to issue transactions against several ledgers, similarly to some notary scheme solutions. The interoperability is enabled through a set of interoperability validators, which are participants from the source and target blockchains. Such validators collect cross-chain transaction requests, sign and deliver them.
Blockchain Agnostic Protocols (including Blockchain of Blockchains)
Zamyatin et al. prove that “there exists no asynchronous CCC [cross-chain communication] protocol tolerant against misbehaving nodes”. The authors use a reduction to the fair exchange problem to prove that correct crosschain communication is as hard as the fair exchange problem. As a consequence of the presented theorem, the authors state that “there exists no CCC protocol tolerant against misbehaving nodes without a trusted third party”. A trusted third party can be centralized or decentralized
Centralized trusted parties are, for example, trusted validators. A decentralized trusted party can be another blockchain, in which their participants agree on the global ledger state via a consensus algorithm. However, the trusted party has to ensure that most participants are honest, guaranteeing the correctness of the process is guaranteed. Cross-chain protocols, therefore “use the consensus of the distributed ledgers as an abstraction for a trusted third party.” ~A Survey on Blockchain Interoperability: Past, Present, and Future Trends
Validators, aka relayers?
Until recently, blockchains did not consider the need for interoperability, as each one focused on resolving specific challenges, leading to data and value silos
...a user should be able to transfer their assets from a blockchain to another or build cross-blockchain decentralized applications.
...has the potential to solve some of the biggest blockchain research challenges. In particular interoperability promotes blockchain scalability, as it provides a way to offload transactions to other blockchains, e.g., via sharding it can promote privacy (by allowing the end-user to use different blockchain for data objects with different privacy requirements), and creates new business opportunities.