Firstly, I hope we agree on the following excerpt taken from the Ethereum wiki:

https://github.com/ethereum/wiki/wiki/Sharding-FAQ
“This sounds like there’s some kind of scalability trilemma at play. What is this trilemma and can we break through it?
The trilemma claims that blockchain systems can only at most have two of the following three properties:
Decentralization (defined as the system being able to run in a scenario where each participant only has access to O(c) resources, ie. a regular laptop or small VPS)
Scalability (defined as being able to process O(n) > O(c) transactions)
Security (defined as being secure against attackers with up to O(n) resources)”

I chose not to include permissioned/private blockchains as they inherently are a tradeoff of Scalability & Confidentiality/Security for Decentralization. This does not seem like a solution to me, rather than a workaround.

The statement that Blockchains scale because they do not require Proof of Work (PoW) is not entirely accurate. Private chains /ican scale because they have a more relaxed threat model than public ones and make assumptions about their participants’ incentives, requiring less effort to secure. Because of that, they can stir away from the resource intense Proof of Work algorithm. Still though, a consensus algorithm that stops malicious actors from interfering with the network’s functionality is still needed (keywords: consensus algorithm, byzantine fault tolerance)

If I may, let me suggest this article from Vitalik Buterin’s blog:
https://blog.ethereum.org/2015/08/07/on-public-and-private-blockchains/