How can the quantitative easing (QE) programme launched in March 2015 by the ECB be successful in the Eurozone (EZ)? What will be its impact on the member countries? And how will it relate to countries' fiscal policies? To address these questions, we use a simple extension of the three-equation New Keynesian model. We modify the benchmark model in two respects: 1) we (re)-introduce an LM money supply and demand equation to capture the fact that the ECB operates at the zero lower bound and hence cannot use a standard Taylor rule; and 2) we extend the model to a two-country framework. The model supports the ECB official view that the channel whereby QE is meant to operate is the reversal of deflationary expectations. It also highlights that instrumental to this goal is the elimination of persistent output gaps, both at the EZ and at the country level, and hence the reduction of country-specific interest-rate spreads - the "unofficial" objective of the programme. We show that QE, if large enough, can succeed for the EZ as a whole. The ECB nevertheless cannot also close individual countries' output gaps, unless specific and unrealistic conditions are met. In this case fiscal accommodation at the country level should also intervene. We show that QE can enhance the effectiveness of fiscal policy, and therefore conclude that the coordination of fiscal and monetary policies is of paramount importance.