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Background

Recently I was reading the article "Quantum Bitcoin: An Anonymous and Distributed Currency Secured by the No-Cloning Theorem of Quantum Mechanics" which demonstrates how a quantum bitcoin could function. The article's conclusion states that:

quantum bitcoins are atomic and there is currently no way to subdivide quantum bitcoin into smaller denominations, or merge them into larger ones.

As there is currently no way to subdivide or merge quantum bitcoins, you can not make change in a transaction. However, I could not understand why subdivision of a quantum bitcoin is not possible.

Question

Why can you not subdivide a quantum bitcoin?

Definitions

A quantum bitcoin - like a regular bitcoin - is a currency with no central authority.

The main idea behind the implementation of a quantum bitcoin is the no-cloning theorem. The no-cloning theorem demonstrates how it is impossible to copy the arbitrary quantum state $ \left| \varphi \right> $.

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  • $\begingroup$ Good question +1. If possible add the definition of "quantum-bitcoin". Secondly, always link to the abstract of a paper, rather than the pdf (I've edited the link, now). $\endgroup$ – Sanchayan Dutta Mar 25 '18 at 16:32
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    $\begingroup$ @Blue OK, I've updated the question and added the definition. I also added information on the no-cloning theorem as it is the main idea behind quantum bitcoins. $\endgroup$ – Daniel Burkhart Mar 25 '18 at 17:00
  • $\begingroup$ Hmm, I do think this would be purely theoretical. I mean, all those qubits are very expensive. A single qubit-coin has to be very expensive to be more expensive than the qubit on which it is created! $\endgroup$ – Discrete lizard Mar 25 '18 at 17:37
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    $\begingroup$ @Discretelizard I'd argue more longterm than theoretical. $\endgroup$ – heather Mar 25 '18 at 17:40
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    $\begingroup$ @heather That's a matter of semantics and optimism. I for one, don't see qubits becoming cheap in the next 50 years. But, who knows. $\endgroup$ – Discrete lizard Mar 25 '18 at 18:03
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Why can you not subdivide a quantum bitcoin?

Anyone can create a Cryptocurrency, how it works is up to them, how well it is received is up to the public, generally it is decided by: Utility, Scarcity, Perceived Value.

As of today a Bitcoin is worth USD 7,073.54, A Bitcoin is 10$^8$ Satoshis which are 0.00000001 Bitcoins, so a Satoshi is worth: 7,073.54 * 0.00000001 = 7.07354 × 10$^{-5}$ USD or 0.00707354 pennies. In total there can be 21 million bitcoin units (2.1 quadrillion Satoshis). The creators of Bitcoin chose that it would be divided into Satoshis.

In Jogenfors' paper, which you cited, he decided that the Quantum Bitcoin protocol (not to be confused with the Qubitcoin (Q2C) a CPU and GPU based Cryptocurrency) will use the no-cloning theorem of quantum mechanics to construct quantum shards and two blockchains.

The answer to your question is, according to section 4.4 - Preventing the Reuse Attack:

"With Bitcoin the blockchain records all transactions and a miner therefore relinquishes control over the mined bitcoin as soon as it is handed over to a recipient. In Quantum Bitcoin, however, there is no record of who owns what, so there is no way to distinguish between the real and counterfeit quantum bitcoin.

We prevent the reuse attack by adding a secondary stage to the minting algorithm, where data is also appended to a new ledger $\mathcal{L^′}$.

...

Quantum shard miners create and sell the quantum shards on a marketplace, another miner (called a quantum bitcoin miner) purchases $m$ quantum shards ${(s, ρi, σi)}1≤i≤m$ on the marketplace that, for all $1 ≤ i ≤ m$, fulfill the following conditions:

  • $\mathsf{Verify}\mathcal{_M} ((s, ρi, σi))$ accepts

  • The timestamp $T$ of the quantum shard in the Quantum Shard ledger $\mathcal{L}$ fulfills $t − T ≤ T_{max}$, where $t$ is the current time". See section "A.2 The Reuse Attack" for further proof.

Shards have a lifetime of $T_{max}$.

While a shard is designed to have a short lifetime the Quantum Bitcoin is designed to have a great longevity (as long as it is intact, undivided), see section "A.3 Quantum Bitcoin Longevity":

"Theorem 4 (Quantum Bitcoin Longevity) The number of times a quantum bitcoin can be verified and reconstructed is exponentially large in $n$.

Proof Corollary 1 shows that the completeness error $\varepsilon$ of $\mathcal{Q}$ is exponentially small in $n$. When verifying a genuine quantum bitcoin \$, the verifier performs the measurement $\mathsf{Verify}\mathcal{_Q}$(\$) on the underlying quantum states $ρ$, which yields the outcome “Pass” with probability $1 − \varepsilon$. Then lemma 2 shows that we can recover the underlying quantum states $\widetilde{p}_{i}$ of \$ so that $ \Vert \widetilde{p}_{i} − p_i \Vert _{tr} \le \sqrt\varepsilon$. As $\varepsilon$ is exponentially small in $n$, the trace distance becomes exponentially small in $n$ as well.

Each time such a quantum bitcoin is verified and reconstructed, the trace distance between the “before” and “after” is exponentially small in $n$. Given any threshold after which we consider the quantum bitcoin “worn out”, the number of verifications it survives before passing this threshold is exponential in $n$.

Theorem 4 shows that a quantum bitcoin \$ can be verified and re-used many times before the quantum state is lost (assuming the absence of noise and decoherence). This is of course analogous to traditional, physical banknotes and coins which are expected to last for a large enough number of transactions before wearing out.".

So, much like paper money, you need the whole bill (blockchain) undivided. Of course, that doesn't prevent you from exchanging one Quantum Bitcoin for however many Bitcoins and then offering Bitcoins (or fiat) as change.

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