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Quantum privacy comparison(QPC) plays an important role in secret ballot elections, private auctions and so on. To date, many multi-party QPC(MQPC) protocols have been proposed to compare the equality of $k(k\geq 3)$ participants. However, there are few examples of MQPC used to compare the sizes or values of their privacies. In this paper, we propose a MQPC protocol by which any $k(k\geq 3)$ participants can compare the sizes of their privacies with executing the protocol just once. The proposed MQPC protocol takes the $d-level$ GHZ states as quantum resources, and a semi-honest $TP$ is introduced to help the participants to determine the relationship of their privacies. Further more, only single-particle unitary transformations and measurements are involved, and the participants need not to share common secrets with each other beforehand which makes the proposed protocol much more efficient. Analysis shows that our protocol is secure against internal and external attack in theory., Comment: 13 pages

The tremendous development of cloud computing and network technology makes it possible for multiple people with limited resources to complete a large-scale computing with the help of cloud servers. In order to protect the privacy of clients, secure multiparty computation plays an important role in the process of computing. Recently, Clementi et al[\textcolor[rgb]{0.00,0.07,1.00}{Phys. Rev. A {\bf 96}, 062317(2017)}] proposed a secure multiparty computation protocol using quantum resources. In their protocol, utilizing only linear classical computing and limited manipulation of quantum information, a method of computing $n-variable$ symmetric Boolean function $f(x_1, x_2, \cdots, x_n)$ with degree 2 is proposed, and all clients can jointly compute $f(x_1, x_2, \cdots, x_n)$ without revealing their private inputs with the help of a sever. They proposed an open problem: are there more simple nonlinear functions like the one presented by them that can be used as subroutines for larger computation protocols? We will give the answer to this question in this paper. Inspired by Clementi et al's work, we continue to explore the quantum realization of Boolean functions. First, we demonstrate a way to compute a class of $n-variable$ symmetric Boolean function $f_n^k$ by using single-particle quantum state $|0\rangle$ and single-particle unitary operations $U_k$. Second, we show that each $n-variable$ symmetric Boolean function can be represented by the linear combination of $f_n^k(k=0,1,\cdots,n)$ and each function $f_n^k(2\leq k\leq n)$ can be used to perform secure multiparty computation. Third, we propose an universal quantum implementation method for arbitrary $n-variable$ symmetric Boolean function $f(x_1, x_2, \cdots, x_n)$. Finally, we demonstrate our secure multiparty computation protocol on IBM quantum cloud platform.