The charge density wave (CDW) is a condensate that often forms in layered materials. It is known to carry electric current \emph{en masse}, but the transport mechanism remains poorly understood at the microscopic level. Its quantum nature is revealed by several lines of evidence. Experiments often show lack of CDW displacement when biased just below the threshold for nonlinear transport, indicating the CDW never reaches the critical point for classical depinning. Quantum behavior is also revealed by oscillations of period $h/2e$ in CDW conductance vs. magnetic flux, sometimes accompanied by telegraph-like switching, in $\text{TaS}_3$ rings above 77 K. Here we discuss further evidence for quantum CDW electron transport. We find that, for temperatures ranging from 9 to 474 K, CDW current-voltage plots of three trichalcogenide materials agree almost precisely with a modified Zener-tunneling curve and with time-correlated soliton tunneling model simulations. In our model we treat the Schrödinger equation as an emergent classical equation that describes fluidic Josephson-like coupling of paired electrons between evolving topological states. We find that an extension of this \lq classically robust' quantum picture explains both the $h/2e$ magnetoconductance oscillations and switching behavior in CDW rings. We consider potential applications for thermally robust quantum information processing systems.