Summary: We prove that the depth of any arithmetic network for deciding membership in a semialgebraic set \(\Sigma \subset \mathbb{R}^{n}\) is bounded from below by \[ c_1 \sqrt{ \frac{\log ({\mathrm b}(\Sigma))}{n}} -c_2 \log n, \] where \({\mathrm b}(\Sigma)\) is the sum of the Betti numbers of \(\Sigma\) with respect to “ordinary” (singular) homology, and \(c_1\), \(c_2\) are some (absolute) positive constants. This result complements the similar lower bound in [J. L. Montaña et al., Appl. Algebra Eng. Commun. Comput. 7, No. 1, 41–51 (1996; Zbl 0844.68069)] for locally closed semialgebraic sets in terms of the sum of Borel-Moore Betti numbers. { }We also prove that if \(\rho: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n-r}\) is the projection map, for some \(r=0, \dots, n\), then the depth of any arithmetic network deciding membership in \(\Sigma\) is bounded by \[ \frac{c_1\sqrt{\log ({\mathrm b}(\rho(\Sigma)))}}{n} - c_2 \log n \] for some positive constants \(c_1\), \(c_2\).