Light Client Support allows for running a node that does not sync the full chain, but rather downloads and verifies only the small parts necessary to send transactions and read state.
Proof-of-Stake replaces Proof-of-Work which opens up options for things like parallelization.
Sharding is a plan to split the blockchain into many smaller chains that are all linked but operate largely independently, allowing for horizontal scaling of the network. Transactions within a shard are synchronous. Transactions across shards are asynchronous.
The next chunk will have the first line to the previous hash, showing data continuity.
Yeah, it's more complicated than that, but that's the raw essential basics.
I think the likely evolution in the future is that most people won't store the blockchain, but that there will be multiple separate currencies and enthusiasts will store the blockchain for the ones they use or are interested in.
EIP 105 (Serenity): Binary sharding plus contract calling semantics
https://github.com/ethereum/EIPs/issues/53
Very technical open discussion here:
https://gitter.im/ethereum/research
I believe they also have weekly Google Hangout on Mondays.
There are other approaches as well. Most actors don't need all past transactions- just the valid spendable outputs. Old txs could be pruned. You can also apply ideas from regular databases, like sharding.
A more difficult task is making it so nodes don't have to see all the ongoing transactions. Ethereum is attempting that with a sharding scheme.