Let us call the current Bitcoin System Bitcoin 1.0 and the sidechain Bitcoin 2.0 So one would take one unit of Bitcoin 1.0 and send it to an unspendable address (e.g. 1111111111111111111114bRaS3) they’d also submit cryptographic proof of the transaction signed by the same private key that sent the transaction as a transaction into Bitcoin 2.0. The protocol of Bitcoin 2.0 would entitle the user to receive one unit of Bitcoin 2.0  This is called “One-way Pegging” as the value of one Bitcoin 2.0 is equal to one Bitcoin 1.0.  This system is only one way and creates a wormhole by which Bitcoin 1.0 disappears as there is no way of getting it back.
I have a hard time swallowing that Bitcoin “isn’t a ledger”. That’s like saying “Bitcoin isn’t the blockchain”, and if you take the blockchain away from Bitcoin, you aren’t really left with much (including, sidechains). Perhaps Bitcoin isn’t a ledger *from the perspective* of individual transactions, but by the same logic, nothing that isn’t transaction data is.
Alpha functions as a sidechain to Bitcoins testnet. The peg mechanism currently works through a centralized protocol adapter, as stated in the sidechains whitepaper. An auditable federation of signers manages Testnet coins transferred to the sidechain. The federation is also relied upon to produce blocks through the signed blocks element. This creates the possibility of exploring the possibilities of the new chain using different security trade-offs.
Public blockchains are open, and therefore are likely to be used by very many entities and gain some network effects. To give a particular example, consider the case of domain name escrow. Currently, if A wants to sell a domain to B, there is the standard counterparty risk problem that needs to be resolved: if A sends first, B may not send the money, and if B sends first then A might not send the domain. To solve this problem, we have centralized escrow intermediaries, but these charge fees of three to six percent. However, if we have a domain name system on a blockchain, and a currency on the same blockchain, then we can cut costs to near-zero with a smart contract: A can send the domain to a program which immediately sends it to the first person to send the program money, and the program is trusted because it runs on a public blockchain. Note that in order for this to work efficiently, two completely heterogeneous asset classes from completely different industries must be on the same database - not a situation which can easily happen with private ledgers. Another similar example in this category is land registries and title insurance, although it is important to note that another route to interoperability is to have a private chain that the public chain can verify, btcrelay-style, and perform transactions cross-chain.
– The manipulation of the blockchain: It is indeed possible to come back at any time on the transactions that have already been added to the blockchain and therefore change the balance of the members. In a public blockchain, such operation would require that 51% of the hashing power (i.e capacity to mine) is concentrated in the hands of the same entity. This not theory anymore since it happened beginning 2014 when the cooperative of GHash minor reached the 51% threshold.

The paper outlines some critical developments and associated problems that were both currently trending and forward-thinking at the time, many of them still very much relevant today. At the time, altcoins were quickly gaining prominence and the problems associated with their volatility, security, and lack of interoperability with Bitcoin raised concerns. The paper primarily addressed 6 issues that pegged sidechains aimed to provide a solution:
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. The advantages of public blockchains generally fall into two major categories:
Liquid is the world's first federated sidechain that enables rapid, confidential, and secure bitcoin transfers. Participating exchanges and Bitcoin businesses deploy the software and hardware that make up the Liquid network, so that they can peg in and out of the Bitcoin blockchain and offer Liquid’s features to their traders. Liquid provides a more secure and efficient system for exchange-side bitcoin to move across the network.

In private blockchains, only specific, pre-chosen entities have the ability to create new transactions on the chain (this is known as “write permissions”). Thus, a private blockchain is a closed network that offers constituents the benefits of the technology, but is not necessarily decentralized or distributed, even among its members. The extent to which each constituent can view (“read”) and create and validate transactions (“write”) is up to the developers of the chain.
The immense promise and accelerated development of permissioned blockchain technology, combined with intense business interest from a wide range of industries, is acting as a perfect stimulant for more and more enterprises to start rolling out blockchain networks into production. I envision these permissioned networks will soon directly or indirectly influence every facet of human enterprise.