Many proteins that undergo sequential enzymatic modification in the Golgi cisternae are displayed at the plasma membrane as cell identity markers. The modified proteins, called glycans, represent a molecular code. The fidelity of this glycan code is measured by how accurately the glycan synthesis machinery realizes the desired target glycan distribution for a particular cell type and niche. In this article, we construct a simplified chemical synthesis model to quantitatively analyse the trade-offs between the number of cisternae, and the number and specificity of enzymes, required to synthesize a prescribed target glycan distribution of a certain complexity to within a given fidelity. We find that to synthesize complex distributions, such as those observed in real cells, one needs to have multiple cisternae and precise enzyme partitioning in the Golgi. Additionally, for a fixed number of enzymes and cisternae, there is an optimal level of specificity (promiscuity) of enzymes that achieves the target distribution with high fidelity. The geometry of the fidelity landscape in the multidimensional space of the number and specificity of enzymes, inter-cisternal transfer rates, and number of cisternae provides a measure for robustness and identifies stiff and sloppy directions. Our results show how the complexity of the target glycan distribution and number of glycosylation enzymes places functional constraints on the Golgi cisternal number and enzyme specificity. Editor's evaluation. This article contributes to an important and largely unexplored topic in cell biology: the understanding of glycosylation. The authors introduce a mathematical model of glycosylation in the Golgi apparatus and use the model to investigate how the complexity (diversity) and fidelity of the plasma membrane glycan distribution depend on parameters such as the number of Golgi cisternae or enzyme specificity. The article is well written and makes the effort to present a rather complex topic in an accessible way by leaving some of the details in the appendices.