how to have synchronization and kind of "time locks" in cryptocurrency?

Hello. I try to visualize a cryptocurrency without blockchain. This is the scenario.
Every peer has a ledger which is a dictionary (think of python) containing public keys and balances. Mister A wants to send 0.5 coins to mister B. Takes a copy of the ledger, makes the appropriate change, signs it and sends it to all the peers. Each peer checks for the validity of the transcaction and if he/she agrees signs the transaction and sends it back to Mister A. Mister A collects ALL the received signed data, signs it again and sends it to every peer in the network. Each peer now checks if the received data is signed by more than 50% of the peers and if yes refreshes his ledger. In a similar manner the new "addresses" are created and added in the ledger. Actually i've thought that proof of work could be used for a new valid address (public key) to be added to the ledger. For example a newcomer would have to do PoW with the last added public key on the ledger as data to be hashed, and then provide it in order to get his public key in the actual ledger and get 1 coin (like what happens when mining in bitcoin). And this with the democratic method mentioned above for the transactions.
But how to provide synchronization in this case? Could there be some kind of "lock" that locks the network for some time to give time for the peer to commit his transaction? Is there an algorithm that takes advantage of the asymmetric encryption and provides synchronization? Do you have any links for me to read that would give me any ideas?
Thanks.