## 4 The GHZ\({}_{3}\)MQPEC Protocol  
Our aim in the current section is to provide a thorough presentation of the new protocol. The protocol is designed to accomplish a quantum private equality comparison among an arbitrarily large number \(n\) of millionaires using only \(|GHZ_{3}\rangle\) states. From now on, we shall  
Figure 1: This figure visualizes the fact that in every triplet of quantum registers \((TIR_{i},SIR_{i},AIR_{i})\), the \(m\) qubits in the corresponding positions belong to the same \(|GHZ_{3}\rangle\) triplet. To make this point clear, they are drawn with the same color.  
refer to it simply by its acronym GHZ\({}_{3}\)MQPEC. Its most important characteristic traits are outlined below:  
* Ease of implementation. Besides EPR pairs, \(|GHZ_{3}\rangle\) triplets are arguably the easiest maximally entangled states that can be produced by a modern quantum apparatus. The fact that, irrespective of the number \(n\) of millionaires, the GHZ\({}_{3}\)MQPEC protocol can be executed with \(|GHZ_{3}\rangle\) triplets exclusively guarantees the feasibility of its implementation.
* Efficiency and scalability. The forthcoming mathematical analysis in Section 6 corroborates the efficiency of the GHZ\({}_{3}\)MQPEC protocol. Moreover, one of its most important qualitative features is that the required quantum resources scale _linearly_ both in the number \(n\) of millionaires and the number \(m\) of qubits necessary to store their fortunes, which further enhances its potential applicability.
* Parallel or sequential execution. Although the execution of the quantum part of the GHZ\({}_{3}\)MQPEC protocol is envisioned to take place completely in parallel, it is also possible to be implemented sequentially. So, if the quantum resources do not suffice for the execution of the protocol in one go, it is possible to partition the \(n\) of millionaires into smaller groups and process these groups sequentially, even to the point of executing the protocol serially for one millionaire at a time.  
Box 1 gives a succinct outline of the intuition behind the GHZ\({}_{3}\)MQPEC protocol, and Box 2 explains how GHZ\({}_{3}\)MQPEC can be simplified in the special case of 2 millionaires. A useful overview of the GHZ\({}_{3}\)MQPEC protocol is also given by the block diagram of Figure 2, containing the actions taking place during the execution of the protocol. Our protocol consists of an initial quantum part followed by a classical part. The former takes place through quantum channels and the latter through public classical channels. Table 1 below contains the acronyms that appear in Figure 2.  
**Box 1.** The intuition behind the GHZ\({}_{3}\)MQPEC protocol.  
The GHZ\({}_{3}\)MQPEC protocol uses entanglement. It is the phenomenon of entanglement that allows for the secret and untraceable encoding of information in the global state of the composite quantum system. The precise mathematical formulation of the correlations among the individual subsystem as encoded into the global state takes the form given by Equation (17), which we call the **Hadamard Entanglement Property**.  
The other prevalent idea is that instead of having [PERSON]\({}_{i}\) and [PERSON]\({}_{j}\) directly comparing their fortunes \(\mathbf{f}_{i}\) and \(\mathbf{f}_{j}\), \(0\leq j\
eq i\leq n-1\), they can both compare their fortune with a fixed but secret number \(\mathbf{s}\). If \(\mathbf{s}\oplus\mathbf{f}_{i}=\mathbf{s}\oplus\mathbf{f}_{j}\), then \(\mathbf{f}_{i}=\mathbf{f}_{j}\), whereas if \(\mathbf{s}\oplus\mathbf{f}_{i}\
eq\mathbf{s}\oplus\mathbf{f}_{j}\), then \(\mathbf{f}_{i}\
eq\mathbf{f}_{j}\). [PERSON] is responsible for randomly choosing a secret number \(\mathbf{s}\) and embedding it into the global entanglement. It goes without saying that [PERSON] must never reveal her secret number \(\mathbf{s}\).