The state is stored in memory and managed through consensus. In Motoko, it’s as simple as assigning it to a stable variable. Calls simply are not transactions, and there won’t be a list of transactions for your canister unless you choose to add one for your particular use case.
Like Ori mentioned, the Internet Computer model is new technology, and has different architecture from existing blockchains. You can just write application logic with our tooling and run it on the Internet Computer. We abstract the consensus logic of the replica out for you. Definitely look forward to the Genesis talks and take a look through our documentation if you’re interested in the project!