Applying to state-side Graduate programs with European degree(s)
Usually a BSc would suffice. This is because PhDs in the US typically contain 2-3 years of lectures, coursework and exams before you do any research. This is the equivalent of doing a separate Master's (and indeed, many students pick up a Master's degree on the way to the PhD, almost by default). In the European system, PhD students start research from day one, so most will have a Master's degree already.

You could consider emailing the graduate admissions people at the universities you're interested in, to find out more about entry requirements specific to your own case.