The devil's staircase is a fractal structure that characterizes the ground state of one-dimensional classical lattice gases with long-range repulsive convex interactions. Its plateaus mark regions of stability for specific filling fractions which are controlled by a chemical potential. Typically such staircase has an explicit particle-hole symmetry, i.e., the staircase at more than half-filling can be trivially extracted from the one at less than half filling by exchanging the roles of holes and particles. Here we introduce a quantum spin chain with competing short-range attractive and long-range repulsive interactions, i.e. a non-convex potential. In the classical limit the ground state features generalized Wigner crystals that -depending on the filling fraction -are either composed of dimer particles or dimer holes which results in an emergent complete devil's staircase without explicit particle-hole symmetry of the underlying microscopic model. In our system the particle-hole symmetry is lifted due to the fact that the staircase is controlled through a two-body interaction rather than a one-body chemical potential. The introduction of quantum fluctuations through a transverse field melts the staircase and ultimately makes the system enter a paramagnetic phase. For intermediate transverse field strengths, however, we identify a region, where the density-density correlations suggest the emergence of quasi long-range order. We discuss how this physics can be explored with Rydberg-dressed atoms held in a lattice.Introduction.-Systems with long-range interactions can feature intriguing physics that is not necessarily present in their short-range counterparts. For example, classical particles in a one-dimensional (1D) lattice interacting via repulsive infinite-range convex potentials lower their interaction energy by assuming a distribution in space as uniform as possible. Remarkably this property makes the ground state configuration independent of the actual details of the interactions, e.g. the specific power-law of the interaction potential, and leads to the formation of a so-called generalized Wigner crystal [1,2]. Furthermore, the permitted filling fractions of the ground state configuration versus the chemical potential form a fractal curve known as the complete devil's staircase [3].Recent years have seen a great success in emulating many-body systems with long-range interactions using ultracold gases. Here, crystalline structures which were originally studied in the context of solid state physics have shown to be present also in ensembles of cold trapped ions [4], polar molecules [5][6][7] and gases of Rydberg atoms [8][9][10][11]. In particular Rydberg gases have witnessed a recent experimental breakthrough in which the adiabatic preparation of a crystalline state and the onset of a staircase structure were shown [12]. A case currently much less studied is when long-range interactions feature competing attraction and repulsion. Systems with such interactions are known to exhibit intricate behavior and in f...