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