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In this paper, by using d-level single-particle states, two novel multi-partyquantum private comparison protocols for size relation comparison with twosemi-honest third parties and one semi-honest third party are constructed,respectively. Here, each protocol can compare the size relation of secretintegers from n parties rather than just the equality within one timeexecution. In each protocol, every third party is assumed to be semi-honest inthe sense that she may misbehave on her own but is not allowed to collude withanyone else; and each party employs the qudit shifting operation to encode hersecret integer. Each protocol can resist both the outside attack and theparticipant attack. Specially, each party's secret integer can be kept unknownto other parties and the third parties. The proposed protocol with two thirdparties is workable in a stranger environment, as there are no communicationand no pre-shared key between each pair of party. The proposed protocol withone third party is workable in an acquaintance environment, as all parties needto share a common private key beforehand.

Multi-party quantum private comparison of size relation with d-level single-particle states