Table 2 Comparison of the proposed MSQPC protocol with the present SQPC protocols of size relationship
|  | Quantum resources | Number of users | Number of TPs | TPs’ knowledge about the comparison results | Type of TPs | Usage of quantum entanglement swapping | Usage of pre-shared key | Usage of unitary operations | Quantum measurements from semiquantum parties | Quantum measurements from quantum parties | Qudit efficiency |
|---|---|---|---|---|---|---|---|---|---|---|---|
Ref. [51] | d-dimensional Bell states | 2 | 1 | Yes | Semi-honest | No | Yes | No | d-dimensional single-particle measurements | d-dimensional Bell state measurements and d-dimensional single- particle measurements | \(\frac{1}{50}\) |
Ref. [52] | d-dimensional GHZ states | 2 | 1 | Yes | Semi-honest | No | Yes | No | d-dimensional single-particle measurements | d-dimensional GHZ state measurements, d-dimensional Bell state measurements and d-dimensional single-particle measurements | \(\frac{1}{42}\) |
The first protocol of Ref. [53] | d-dimensional single-particle states | 2 | 1 | Yes | Semi-honest | No | Yes | No | d-dimensional single-particle measurements | d-dimensional single-particle measurements | \(\frac{1}{50}\) |
The second protocol of Ref. [53] | d-dimensional single-particle states | 2 | 1 | Yes | Semi-honest | No | Yes | Yes | d-dimensional single-particle measurements | d-dimensional single-particle measurements | \(\frac{1}{14}\) |
Ref. [54] | d-dimensional Bell states | 2 | 1 | Yes | Semi-honest | No | Yes | No | No | d-dimensional Bell state measurements and d-dimensional single- particle measurements | \(\frac{1}{38}\) |
Ref. [55] | d-dimensional single-particle states | 2 | 1 | No | Semi-honest | No | Yes | No | d-dimensional single-particle measurements | d-dimensional single- particle measurements | \(\frac{1}{51}\) |
The proposed MSQPC protocol | d-dimensional single-particle states | N | 2 | Yes | Semi-honest | No | No | No | d-dimensional single-particle measurements | d-dimensional single- particle measurements | \(\frac{2}{N^{2} + 49N}\) |