Toehold-mediated DNA strand displacement is the foundation of dynamic DNA nanotechnology, encompassing a wide range of tools with diverse functions, dynamics, and thermodynamic properties. However, a majority of these tools are limited to unidirectional reactions driven by thermodynamics. In response to the growing field of dissipative DNA nanotechnology, we present an approach: DNAzyme-based dissipative DNA strand displacement (D-DSD), which combines the principles of dynamic DNA nanotechnology and dissipative DNA nanotechnology. D-DSD introduces circular and dissipative characteristics, distinguishing it from the unidirectional reactions observed in conventional strand displacement. We investigated the reaction mechanism of D-DSD and devised temporal control elements. By substituting temporal components, we designed two distinct temporal AND gates using fewer than 10 strands, eliminating the need for complex network designs. In contrast to previous temporal logic gates, our temporal storage is not through dynamics control or cross-inhibition but through autoregressive storage, a more modular and scalable approach to memory storage. D-DSD preserves the fundamental structure of toehold-mediated strand displacement, while offering enhanced simplicity and versatility.