It's important to distinguish between measures and analyses, because only analyses can be quantitative or qualitative, not measures.
Measures are, essentially, systematic processes by which we acquire our data, and analyses are processes we use to look at the data. As a rule of thumb, the difference is not hard to find and is given in the name: quantitative analysis deals with numbers, and qualitative analysis doesn't. It looks more complex than it is.
ANOVA and t-tests are both forms of statistical linear regression, for example. They 'concentrate' many standardized observations down to a manageable model we can easily deal with, like an extremely effective form of memory chunking. They're quantitative because they have to do with numbers. Analyses can be quantitative or qualitative, and you can easily tell them apart by whether the technique relies on numbers or judgment.
You didn't actually give any examples of qualitative analysis in the question, but the prototypical example of qualitative analysis is the case study. For various reasons, it's sometimes necessary or just useful to conduct a study based on a single case (e.g., the memory patient HM). These studies can draw meaningful and very useful conclusions based on qualitative analyses like comparative studies, double dichotomy, etc., when we would never be able to reach a meaningful level of statistical power (e.g., because only one or very few patients is/are known to exist in the world).
Measures are neither quantitative or qualitative. Analytic techniques are quantitative or qualitative based on whether or not they work through numbers or human judgment. (Sorry about the formatting, I'm posting from a tablet and will clean it up later.)