Molecular circuits crafted from DNA molecules harness the inherent programmability and biocompatibility of DNA to intelligently steer molecular machines in the execution of microscopic tasks. In comparison to combinational circuits, DNA-based temporal circuits boast supplementary capabilities, allowing them to proficiently handle the omnipresent temporal information within biochemical systems and life sciences. However, the lack of temporal mechanisms and components proficient in comprehending and processing temporal information presents challenges in advancing DNA circuits that excel in complex tasks requiring temporal control and time perception. In this study, we engineered temporal logic circuits through the design and implementation of a dual cross-inhibition mechanism, which enables the acceptance and processing of temporal information, serving as a fundamental building block for constructing temporal circuits. By incorporating the dual cross-inhibition mechanism, the temporal logic gates are endowed with cascading capabilities, significantly enhancing the inhibitory effect compared to a cross-inhibitor. Furthermore, we have introduced the annihilation mechanism into the circuit to further augment the inhibition effect. As a result, the circuit demonstrates sensitive time response characteristics, leading to a fundamental improvement in circuit performance. This architecture provides a means to efficiently process temporal signals in DNA strand displacement circuits. We anticipate that our findings will contribute to the design of complex temporal logic circuits and the advancement of molecular programming.