Research progress in quantum computing has, thus far, focused on a narrow set of application domains. Expanding the suite of quantum application domains is vital for the discovery of new software toolchains and architectural abstractions. In this work, we unlock a new class of applications ripe for quantum computing research -- computational cognitive modeling. Cognitive models are critical to understanding and replicating human intelligence. Our work connects computational cognitive models to quantum computer architectures for the first time. We release QUATRO, a collection of quantum computing applications from cognitive models. The development and execution of QUATRO shed light on gaps in the quantum computing stack that need to be closed to ease programming and drive performance. Among several contributions, we propose and study ideas pertaining to quantum cloud scheduling (using data from gate- and annealing-based quantum computers), parallelization, and more. In the long run, we expect our research to lay the groundwork for more versatile quantum computer systems in the future.