  Self-testing of quantum devices based on observed measurement statistics is a
method to certify quantum systems using minimal resources. In Ref. [Phys. Rev.
\textbf{A} 101, 032106 (2020)], a scheme based on observing measurement
statistics that demonstrate Kochen-Specker contextuality has been shown to
certify two-qubit entangled states and measurements without the requirement of
spatial separation between the subsystems. However, this scheme assumes a set
of compatibility conditions on the measurements which are crucial to
demonstrating Kochen-Specker contextuality. In this work, we propose a
self-testing protocol to certify the above two-qubit states and measurements
without the assumption of the compatibility conditions, and at the same time
without requiring the spatial separation between the subsystems. Our protocol
is based on the observation of sequential correlations leading to the maximal
violation of a temporal inequality derived from non-contextuality inequality.
Moreover, our protocol is robust to small experimental errors or noise.
