Crystal graphs are powerful combinatorial tools for working with the plactic monoid and symmetric functions. Quasi-crystal graphs are an analogous concept for the hypoplactic monoid and quasi-symmetric functions. This paper makes a combinatorial study of these objects. A previously-observed isomorphism of components of the quasi-crystal graph is explained and explicitly described using a new combinatorial structure called a quasi-array. Then two conjectures of Maas-Gariépy on the interaction of fundamental quasi-symmetric functions and Schur functions and on the arrangement of quasi-crystal components within crystal components are settled, the former positively, the latter negatively.