The Quantified Reflection Calculus with N modalities, or QRC_N, is a quantified and strictly positive modal logic inspired by the provability reading of the modal symbols. The non-modal language consists of a verum constant and relational symbols as atomic formulas, conjunctions, and universal quantifiers. The available modal connectives are labeled diamonds, one for each natural number n < N. QRC_N statements are assertions of the form A \vdash B, with A and B in this strictly positive language. We describe the axiomatic system for QRC_N and see that it is sound with respect to constant domain Kripke models. Then we focus on the QRC_1 fragment, which is complete with respect to finite and constant domain Kripke models, and thus decidable. The completeness and decidability of the general case with N modalities is still an open problem. (Joint work with Joost J. Joosten)