Light clients follow Ethereum by checking signatures from a rotating committee of 512 validators.
The idea: Every 256 epochs (about 27 hours) a random committee of 512 validators is chosen, and its members sign each new block header with BLS, merged into one signature. A light client only tracks the committee and checks those signatures, then uses Merkle proofs to check any data it needs, so it can follow the chain with very little data.
Why it matters: Phones, browsers and bridges use them to trust Ethereum without running a full node.
On the map: sync-committee signatures + state proofs
Light clients follow Ethereum without downloading everything. Every 256 epochs (about 27 hours) a sync committee of 512 validators is chosen[1]. A light client checks their aggregated BLS signature on each header and then verifies data with Merkle proofs into the state.
Many bridges and other chains verify Ethereum the same way. They inherit BLS, the hash tries, and the assumption that most of the sync committee is honest. sync-committee signatures aren't slashable[2], so that rests on the committee being a random sample of mostly honest stake. A light client also has to start from a trusted recent block root[1].
If any of these breaks, this breaks too.
If this breaks, these break with it.
Nothing on the map depends on it: it is an end point, something people use directly.