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Amorphous materials are solids that lack long-range atomic order but possess complex short- and medium-range order. Unlike crystalline materials that can be described by unit cells containing few up to hundreds of atoms, amorphous materials require larger simulation cells with at least hundreds or often thousands of atoms. Inverse design of amorphous materials with probabilistic generative models aims to generate the atomic positions and elements of amorphous materials given a set of desired properties. It has emerged as a promising approach for facilitating the application of amorphous materials in domains such as energy storage and thermal management. In this paper, we introduce AMShortcut, an inference- and training-efficient probabilistic generative model for amorphous materials. AMShortcut enables accurate inference of diverse short- and medium-range structures in amorphous materials with only a few sampling steps, mitigating the need for an excessive number of sampling steps that hinders inference efficiency. AMShortcut can be trained once with all relevant properties and perform inference conditioned on arbitrary combinations of desired properties, mitigating the need for training one model for each combination. Experiments on three amorphous materials datasets with diverse structures and properties demonstrate that AMShortcut achieves its design goals.