How to measure quality of education across countries? For my research, I need a measure of the quality of education that I could apply to multiple countries. It must include quality of tertiary education. 

Standard measures like HDI Education Index are not precise enough. For example in HDI Education Index, Kazakhstan is ranked along Hong Kong, Belarus above Singapore, etc. Such absurd discrepancies destroy any hope for good results for my model. 

Please, if you have an idea (no matter how wild it is) what I can use as a proxy for quality of education, please share it.
A good reference is the Barro-Lee Dataset wich gives educational attainment
for 146 countries from 1960 to 2010. The data are disaggregated by sex and by 5-year age intervals. Their estimates of educational attainment provide a reasonable proxy for the stock of human capital.

An older reference is hosted by the World Bank.

In case you need more recent data, you may apply their methodology.