The general Tannaka duality in part says that a (quasi-compact, quasi-separated) scheme can be recovered from the tensor category of its quasi-coherent sheaves. Jacob Lurie's paper Tannaka Duality for Geometric Stacks even extends it to certain relative situations involving morphisms of stacks.
This gives rise to the question: if we think of a scheme $X$ as its category of quasi-coherent sheaves $QC_X$, how can we think of the motives associated to $X$ in the same terms? Is there a construction of motives starting with $QC_X$? The reason for asking is that both motives and sheaves are linearizing devices, so the connection between the two might be easier to see.