The higher valuations of private companies IMO has little to do with quantitative easing. Despite any bubble talk, very little funding actually goes into these private companies. The S&P 500 returns nearly 1 trillion a year to shareholders via dividends, VCs put like 50 billion into these high growth startups.
Low interest rates from quantitative easing might contribute greatly, but I wouldn't say the inflation targets have much to do with it.
Higher valuations come partly from businesses having more cash because they are not spending money expanding, and because the low interest rates encourage companies to borrow money to buy back stock, which increases EPS and dividends, which increases share prices. Of course, there has been some legitimate growth over the past years, too.
[1] http://www.usinflationcalculator.com/inflation/current-infla...
It additionally devalues the dollar, which leads to assets to be worth bigger numbers. QE has been mainly flowing to investors and the financial sector, and ts been staying there, hence not translating to consumer price inflation.
So yes QE is responsible and QE caused inflation and inflation is also responsible. This inflation is confined to investment assets.