Vision-Language Models (VLMs) achieved strong performance on a variety of tasks (e.g., image-text retrieval, visual question answering). However, most VLMs rely on coarse-grained image-caption pairs for alignment, relying on data volume to resolve ambiguities and ground linguistic concepts in images. The richer semantic and syntactic structure within text is largely overlooked. To address this, we propose HIerarchically STructured Learning (HIST) that enhances VLM training without any additional supervision, by hierarchically decomposing captions into the constituent Subject, Noun Phrases, and Composite Phrases. Entailment between these constituent components allows us to formulate additional regularization constraints on the VLM attention maps. Specifically, we introduce two novel loss functions: (1) Subject Loss, which aligns image content with the subject of corresponding phrase, acting as an entailment of standard contrastive/matching losses at the Phrase level; (2) Addition Loss, to balance attention across multiple objects. HIST is general, and can be applied to any VLM for which attention between vision and language can be computed; we illustrate its efficacy on BLIP and ALBEF. HIST outperforms baseline VLMs, achieving up to +9.8% improvement in visual grounding, +6.3% in multi-object referring segmentation, +1.1% in image-text retrieval, and +0.2% in visual question answering, underscoring the value of structuring learning in VLMs.