You write "qualitative easing", but I think you refer to quantitative easing. I'll just do both.
Quantitative easing corresponds to the central bank (CB) expanding its balance sheets by "buying" assets. This is typically done in secondary markets. It mainly injects liquidity into the system. To the extend that there is an additional buyer of assets now, the price of assets/investment (interest rates) decreases. However, the quantitative impact should be negligible: Through its demand, the CB increases value and liquidity in the markets it is operating. The scale of its operations should be too small however to affect the aggregate interest rate.
Qualitative easing is a relatively new expression and refers to the riskiness of the stocks that the CB is investing into. In contrast to quantitative easing, which is about the magnitude of assets on the CB's balance sheets, qualitative easing is about the riskiness on the CB
s balance sheets, and hence the decrease in aggregate risk (on the banks' balance sheets).
Independence of a central bank
When the CB holds assets, it is interested in their value. This may lead it to do policies that go against its primary directive (e.g. inflation stability). Even if it does not do so, effectiveness of a central bank comes from its capacity of controlling expectations. It is sufficient for households/firms to expect the central bank to do "bad" policies in order to decrease the effectiveness of the CB.
Number of Goals
I don't have sources on this, but I seem to remember that central banks with one clear goal (i.e. monetary stability) are more effective than those with a basket of goals (i.e. monetary stability, GDP growth, decrease of unemployment rate). A general criticism can be that QE are not operations that help with the important margin, monetary stability - and that the central bank should focus on that instead.
Note that these are not just esoteric points. In fact, the academics and central bankers at the "Rethinking Macro Policy" conference agreed that (quoting Blanchard)
Throughout the conference, e.g., in Gill Marcus’ talk, and actually throughout the various meetings which took place during the IMF meetings in the following days, policy makers remarked and complained about the heavy burden placed on monetary policy in this crisis, and the danger of a political backlash against central banks. Even as the crisis recedes, it is clear that central banks will end up with substantially more responsibilities—whether they are given in full or shared—for financial regulation, financial supervision, and the use of macro prudential tools.
While even the use of the policy rate has distributional implications, these implications are much more salient in the case of regulation or macro prudential tools, such as the loan-to-value ratio. The general consensus was that these distributional implications could not be ignored, and that while central banks should retain full independence with respect to traditional monetary policy, this cannot be the case for regulation or macro prudential tools.