“Given all of this, it may seem like private blockchains are unquestionably a better choice for institutions. However, even in an institutional context, public blockchains still have a lot of value, and, in fact, this value lies to a substantial degree in the philosophical virtues that advocates of public blockchains have been promoting all along, among the chief of which are freedom, neutrality and openness.” 

“A private blockchain is hardly different from a traditional database. The term is synonymous with glorified databases. But the advantage is that if they are to ever start adding public nodes to it then it becomes so much more. An open blockchain is the best method for having a trustless ledger. The broader the range of decentralized adoption the better. The Bitcoin blockchain hits all those points. 
Peer-to-peer blockchain networks lack centralized points of vulnerability that computer crackers can exploit; likewise, it has no central point of failure. Blockchain security methods include the use of public-key cryptography.[4]:5 A public key (a long, random-looking string of numbers) is an address on the blockchain. Value tokens sent across the network are recorded as belonging to that address. A private key is like a password that gives its owner access to their digital assets or the means to otherwise interact with the various capabilities that blockchains now support. Data stored on the blockchain is generally considered incorruptible.[1]
Public blockchains are also expensive, and not just in terms of money. The time and energy required to process transactions on public chains is more intensive than that of non-public chains. This is because every single node on the chain must authorize each new transaction before it is added to the chain, which requires a large amount of electricity and time (not to mention money).
For example, let’s say we have side chain 1 (SC1) and side chain 2 (SC2). A transaction occurs on SC1. A node in SC1 broadcasts the transaction to nodes in the main chain to record this transaction. The same node of SC1 calls a function from SC2 with a proof. The function in the nodes of SC2 verifies the proof on the main chain. The function gets executed.
The Bitcoin White Paper was published by Satoshi Nakamoto in 2008; the first Bitcoin block got mined in 2009. Since the Bitcoin protocol is open source, anyone could take the protocol, fork it (modify the code), and start their own version of P2P money. Many so-called altcoins emerged and tried to be a better, faster or more anonymous than Bitcoin. Soon the code was not only altered to create better cryptocurrencies, but some projects also tried to alter the idea of blockchain beyond the use case of P2P money.
The witnesses who put more funds in escrow have a greater chance of mining (or minting) the next block. The incentives line up nicely here. There are only a few witnesses and they get paid to be witnesses, so they are incentivized to not cheat. If they do cheat and get caught, they not only get voted out in favor of the next eagerly awaiting witness, they lose all the funds they had in escrow.
Blockchains that are private or permissioned work similarly to public blockchains but with access controls that restrict those that can join the network, meaning it operates like a centralised database system of today that limits access to certain users. Private Blockchains have one or multiple entities that control the network, leading to the reliance on third-parties to transact. A well-known example would be Hyperledger.
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My take is that the Bitcoin architecture is a solution to the problem of how to maintain consensus about a ledger when the participants are unknown and many of them are adversarial (I know this is loose language… computer scientists working in the consensus space are more precise but I think this captures the essence…. i.e. we’re explicitly in a world where there is no “leader” and no identities for those providing the consensus services).

Step back from the details for moment and consider what’s been described.  We now have a way to move coins from Bitcoin onto another platform (a sidechain) and move them back again.   That’s pretty much what we do when we move them to a wallet platform or an exchange.  The difference is that the “platform” they’ve been moved to is also a blockchain… so it has the possibility of decentralised security, visibility and to gain from other innovation in this space.

Developers and Cryptocurrency enthusiasts have been looking at expanding Bitcoins functionality as mainstream adoption increases. Side chains would increase the resilience of Bitcoin: If one of the sidechains was to be compromised, only the Bitcoins on that chain would be lost, while other sidechains and the Blockchain would continue like normal. This would further stabilize the Bitcoin network and increase security.
Sidechains as an idea have existed and had been floating around for quite some time now, the bases is to extend the decentralization of trust into other sectors and to other digital assets. However, while this all sounds great it's a perfect example of good in theory but not so much in practice. Nevertheless, this hasn't stopped people from trying with groups such as Blockstream exploring the idea and our friends over at Rootstock co-creating a Sidechain which is allowing Litecoin and Bitcoin to execute smart contracts and all without changing the core software of the original currency.
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This is what, at its core, state channels are. Imagine we wanted to play a game of Starcraft and have a smart contract that pays 1 ETH to the winner. It would be ridiculous for each participant to have to write on the main Ethereum network each time a Zergling was killed by a Zealot, or when a Command Center was upgraded to an Orbital Command. The gas cost (Ethereum gas, not Starcraft gas) and time for each transaction would be prohibitive.

What is the difference between a public blockchain and a private blockchain? Does it matter? Which is better? Gallactic believes that currently there are pros and cons between both Private and Public Blockchains, but time and “convergence”, a term that is gaining prominence in the Blockchain Industry, is clearly showing that the lines between these categories, once clear, are starting to fade.