Nodes can be trusted to be very well-connected, and faults can quickly be fixed by manual intervention, allowing the use of consensus algorithms which offer finality after much shorter block times. Improvements in public blockchain technology, such as Ethereum 1.0's uncle concept and later proof of stake, can bring public blockchains much closer to the "instant confirmation" ideal (eg. offering total finality after 15 seconds, rather than 99.9999% finality after two hours as does Bitcoin), but even still private blockchains will always be faster and the latency difference will never disappear as unfortunately the speed of light does not increase by 2x every two years by Moore's law.
Now, making experimental or rapid changes to Bitcoin is very risky and so change happens slowly. So if the one-size-fits-all architecture of Bitcoin doesn’t suit a particular use-case, you have a problem. You either have to use an entirely different cryptocurrency (or build one!). Or you have to use (or build) a centralized service, which brings new risks.
Blockstream has also released an “Alpha” sidechain with all of those features up and running except the last, coupled to the Bitcoin testnet. (Used for testing Bitcoin software without putting real value at risk.) In the absence of the Bitcoin protocol change that will cryptographically secure the programmatic transfer of value between Bitcoin and sidechains, they’re cooperating with several external organizations to perform and validate those transfers. If and when that protocol change happens, though, pegged sidechains will be as permissionless, and as decentralized, as Bitcoin itself.

A federation is a group that serves as the intermediary between a parent chain and its corresponding sidechain. It is an additional layer in the protocol but serves a key function and is what Blockstream’s Liquid sidechain uses. Due to the lack of expressiveness of Bitcoin’s scripting language, an externally implemented and mutually distrusting set of members form a federated peg.

This list is not exhaustive. There are plenty of public blockchains, and they are actively adopted by such industries as FinTech, gaming, logistics, and beyond. However, it not always makes sense to move certain processes and businesses to the public network as the latter are characterized by comparatively low speed of transactions execution and high costs. Indeed, every transaction requires a consensus of the entire network. Unfortunately, it takes time and resources.
Eris Industries, aims to be the provider of shared software database using blockchain technology. Blockstack, aims to provide financial institutions back office operations, including clearing & settlement on a private blockchain. Multichain, provider an open source distributed database for financial transactions. Chain Inc., a provider of blockchain API's. Chain partnered with Nasdaq OMX Group Inc., to provide a platform that enables trading private company shares with the blockchain.
Always there is a balance in nature, even in blockchains. If you want to have extra features, you need to make a sacrifice from your current features. For example to have high speed and volume; you need to give some from your security & immutability by doing consensus with smaller groups or you need to use different methods in consensus like POS / PBFT. (Proof of Stake / Practical Byzantine Fault Tolerance)
The first work on a cryptographically secured chain of blocks was described in 1991 by Stuart Haber and W. Scott Stornetta.[10][6] They wanted to implement a system where documents' timestamps could not be tampered with or backdated. In 1992, Bayer, Haber and Stornetta incorporated Merkle trees to the design, which improved its efficiency by allowing several documents to be collected into one block.[6][11]
“Further, contribution is weighted by computational power rather than one threshold signature contribution per party, which allows anonymous membership without risk of a Sybil attack (when one party joins many times and has disproportionate input into the signature). For this reason, the DMMS has also been described as a solution to the Byzantine Generals Problem[AJK05].”
In simple terms, public blockchains can receive and send transactions from anybody in the world. They can also be audited by anybody, and every node has as much transmission power as any other. Before a transaction is considered valid, it must be authorized by each of its constituent nodes via the chain’s consensus process. As long as each node abides by the specific stipulations of the protocol, their transactions can be validated, and thus add to the chain
Public blockchains are just that, public. Anyone that wants to read, write, or join a public blockchain can do so. Public chains are decentralized meaning no one body has control over the network, ensuring the data can’t be changed once validated on the blockchain. Simply meaning, anyone, anywhere, can use a public blockchain to input transactions and data as long as they are connected to the network.