Ethereum keeps every balance and contract's data in a tree of fingerprints, sealed by one root fingerprint in each block.
The idea: Ethereum's whole state is arranged as a tree of hashes: each account's data is hashed, those hashes are grouped and hashed again, up to one root in each block header. To prove your balance to someone who only trusts the header, you hand over the hashes along the path from your account to the root. Faking a different balance would take a Keccak collision.
Why it matters: Lets light clients and bridges prove facts about Ethereum with short proofs.
On the map: Merkle-Patricia tries over Keccak-256
Ethereum's execution state (every account balance, nonce and storage slot), each block's transactions and its receipts are stored in Merkle-Patricia tries hashed with Keccak-256[1]. Their root hashes go in the block header.
A short proof shows that a value is in the state, which is what light clients, bridges and optimistic-rollup fault proofs rely on. Security needs Keccak's collision resistance. The planned binary state tree would replace these tries[2].
If any of these breaks, this breaks too.
If this breaks, these break with it.