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In this paper, we propose NeuralQAAD, a differentiable point cloud compression framework that is fast, robust to sampling, and applicable to high resolutions. Previous work that is able to handle complex and non-smooth topologies is hardly scaleable to more than just a few thousand points. We tackle the task with a novel neural network architecture characterized by weight sharing and autodecoding. Our architecture uses parameters much more efficiently than previous work, allowing us to be deeper and scalable. Futhermore, we show that the currently only tractable training criterion for point cloud compression, the Chamfer distance, performances poorly for high resolutions. To overcome this issue, we pair our architecture with a new training procedure based upon a quadratic assignment problem (QAP) for which we state two approximation algorithms. We solve the QAP in parallel to gradient descent. This procedure acts as a surrogate loss and allows to implicitly minimize the more expressive Earth Movers Distance (EMD) even for point clouds with way more than $10^6$ points. As evaluating the EMD on high resolution point clouds is intractable, we propose a divide-and-conquer approach based on k-d trees, the EM-kD, as a scaleable and fast but still reliable upper bound for the EMD. NeuralQAAD is demonstrated on COMA, D-FAUST, and Skulls to significantly outperform the current state-of-the-art visually and in terms of the EM-kD. Skulls is a novel dataset of skull CT-scans which we will make publicly available together with our implementation of NeuralQAAD.