MQuAcq is an algorithm for active constraint acquisition that has been shown to outperform previous algorithms such as QuAcq and MultiAcq. In this paper, we exhibit two important drawbacks of MQuAcq. First, for each negative example, the number of recursive calls to the main procedure of MQuAcq can be non-linear, making it impractical for large problems. Second, MQuAcq, as well as QuAcq and Multi-Acq, does not take into account the structure of the learned problem. We propose MQuAcq-2, a new algorithm based on MQuAcq that integrates solutions to both these problems. MQuAcq-2 exploits the structure of the learned problem by focusing the queries it generates to quasi-cliques of constraints. When dealing with a negative query, it only requires a linear number of iterations. MQuAcq-2 outperforms MQuAcq, especially on large problems.