Certification of three black boxes with unsharp measurements using $3 \rightarrow 1 $ sequential quantum random access codes

Unsharp measurements play an increasingly important role in quantum information theory. In this paper, we study a three-party prepare-transform-measure experiment with unsharp measurements based on $3 \rightarrow 1$ sequential random access codes (RACs). We derive optimal trade-off between the two correlation witnesses in $3 \rightarrow 1$ sequential quantum random access codes (QRACs), and use the result to complete the self-testing of quantum preparations, instruments and measurements for three sequential parties. We also give the upper and lower bounds of the sharpness parameter to complete the robustness analysis of the self-testing scheme. In addition, we find that classical correlation witness violation based on $3 \rightarrow 1$ sequential RACs cannot be obtained by both correlation witnesses simultaneously. This means that if the second party uses strong unsharp measurements to overcome the classical upper bound, the third party cannot do so even with sharp measurements. Finally, we give the analysis and comparison of the random number generation efficiency under different sharpness parameters based on the determinant value, $2 \rightarrow 1$ and $3 \rightarrow 1$ QRACs separately. This letter sheds new light on generating random numbers among multi-party in semi-device independent framework.