Those are two different problems: scalability and finality.
Obviously every transaction can not be processed and stored by everyone. That much is clear even to casual observers. There has been two or three main ways people have tried to achieve this during the past decade.
The obvious thing to try would be to shard the blockchain like you would a database. This turns out to be hard to do in a trustless way since shards would need to interact. This realization and the contracts required to securely swap assets between otherwise separate chains leads naturally to:
Full on separate blockchains that run in parallel to the main one, checkpointing when needed (rootstock, drivechains). These are not limited by the main chain and can be specialized for custom use cases. The parallel chains are only interoperable by way of the main chain and need not know about each other, which helps scaling out.
Payment channels by the way of time locked contracts. Satoshi sketched out an initial implementation that turned out to be flawed. This has since been improved on and made bidirectional and made into a standard which is now the Lightning network. It has a number of real world limitations but the general idea is that only the parties involved in a transaction needs to know about it. An added benefit of this is that finality among these parties is immediate.
There have also been some work squashing a large number of transactions into a large transaction. This has the added benefit of obfuscating the flow of individual transactions, which otherwise makes everyone's holdings transparent (mimblewimble, grin). This requires new signature schemes and is hard to retrofit to existing blockchains and make security guarantees about.
There used to be ideas about Chaum like schemes on top of blockchains, but most of that interest probably went on into separate blockchain schemes.
Those are some of the ideas that have been tried, most have shown some promise but are more or less still at the research stage. Don't expect radical changes overnight.
You forgot the most important thing wrt scaling: everyone don't need to run a full node (store the block chain).
Most mobile wallets are light wallets, that query servers for the information on demand. It works great, but you have the risk of the server lying to you.
So the next level up is SPV wallets, which verify that transactions are included in blocks and that the proof-of-work is valid. So the cheat them you need to reproduce POW, which is very expensive, and also very secure.
This notion that everyone needs to run a full node is simply false. SPV security, and even light wallets security, is enough for almost everyone. Exchanges, payment processors and the paranoid few can still run full nodes.
Thanks for this. I was wondering about this exact thing. Any idea about the the current transactions/second with finality of the BTC or ETH? What is expected from the new methods?
