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Traditional logical equivalence checking (LEC) which plays a major role in entire chip design process faces challenges of meeting the requirements demanded by the many emerging technologies that are based on logic models different from standard complementary metal oxide semiconductor (CMOS). In this paper, we propose a LEC framework to be employed in the verification process of beyond-CMOS circuits. Our LEC framework is compatible with existing CMOS technologies, but, also able to check features and capabilities that are unique to beyond-CMOS technologies. For instance, the performance of some emerging technologies benefits from ultra-deep pipelining and verification of such circuits requires new models and algorithms. We, therefore, present the Multi-Cycle Input Dependency (MCID) circuit model which is a novel model representation of design to explicitly capture the dependency of primary outputs of the circuit on sequences of internal signals and inputs. Embedding the proposed circuit model and several structural checking modules, the process of verification can be independent of the underlying technology and signaling. We benchmark the proposed framework on post-synthesis rapid single-flux-quantum (RSFQ) netlists. Results show a comparative verification time of RSFQ circuit benchmark including 32-bit Kogge-Stone adder, 16-bit integer divider, and ISCAS'85 circuits with respect to ABC tool for similar CMOS circuits.