Despite the wide success of neural networks, their computational cost is very high. Quantization techniques reduce this cost but can result in changing the classifications of the original floating-point network, even if the training is quantization-aware. In this work, we rely on verification to design correction systems that detect classification inconsistency at inference time and eliminate them. The key idea is to overapproximate the space of inconsistent inputs with their maximal classification confidence. The main challenge in computing this confidence is that it involves analyzing a quantized network, which introduces a very high degree of nonlinearity, over all possible inputs. We propose CoMPAQt, an algorithm for computing this confidence. CoMPAQt relies on a novel encoding of quantization in mixed-integer linear programming (MILP) along with customized linear relaxations to reduce the high complexity. To prune the search space, it ties the computations of the quantized network and its floating-point counterpart. Given this confidence, we propose two correction mechanisms. The first mechanism guarantees to return the classification of the floating-point network and relies on networks with increasing bit precisions. The second mechanism mitigates classification inconsistencies by an ensemble of quantized networks. We evaluate our approach on MNIST, ACAS-Xu, and tabular datasets over fully connected and convolutional networks. Results show that our first correction mechanism guarantees 100% consistency with the floating-point network’s classifications while reducing its computational cost by 3.8x, on average. Our second mechanism reaches an almost perfect consistency guarantee in our experiments while reducing the computational cost by 4.1x. Our work is the first to provide a formal guarantee on the classification consistency of a quantized network.