We study the quantumness of correlations for ensembles of bi- and multi-partite systems and relate it to the task of quantum data hiding. Quantumness is here intended in the sense of minimum average disturbance under local measurements. We consider a very general framework. but focus on local complete von Neumann measurements as the cause of the disturbance. https://allfixelectricales.shop/product-category/thermostat-palnut/