Neurovascular coupling (NVC) is the tight relationship between changes in cerebral blood flow and neural activation. NVC can be evaluated non-invasively using transcranial Doppler ultrasound (TCD)-measured changes in brain activation (cerebral blood velocity [CBv]) using different cognitive tasks and stimuli. This study used a novel approach to analyzing CBv changes occurring in response to 20 tasks from the Addenbrooke's Cognitive Examination III in 40 healthy individuals. The novel approach compared various information entropy families (permutation, Tsallis, and Rényi entropy) and statistical complexity measures based on disequilibrium. Using this approach, we found the majority of the attention, visuospatial, and memory tasks from the Addenbrooke's Cognitive Examination III that showed lower statistical complexity values when compared with the resting state. On the entropy-complexity (HC) plane, a receiver operating characteristic curve was used to distinguish between baseline and cognitive tasks using the area under the curve. Best area under the curve values were 0.91 ± 0.04, p = .001, to distinguish between resting and cognitively active states. Our findings show that brain hemodynamic signals captured with TCD can be used to distinguish between resting state (baseline) and cognitive effort (stimulation paradigms) using entropy and statistical complexity as an alternative method to traditional techniques such as coherent averaging of CBv signals. Further work should directly compare these analysis methods to identify the optimal method for analyzing TCD-measured changes in NVC.