  In this paper we consider realistic model of inflation embedded in the
framework of loop quantum cosmology. Phase of inflation is preceded here by the
phase of a quantum bounce. We show how parameters of inflation depend on the
initial conditions established in the contracting, pre-bounce phase. Our
investigations indicate that phase of the bounce easily sets proper initial
conditions for the inflation. Subsequently we study observational effects that
might arise due to the quantum gravitational modifications. We perform
preliminary observational constraints for the Barbero-Immirzi parameter
$\gamma$, critical density $\rho_{\text{c}}$ and parameter $\lambda$. In the
next step we study effects on power spectrum of perturbations. We calculate
spectrum of perturbations from the bounce and from the joined bounce+inflation
phase. Based on these studies we indicate possible way to relate quantum
cosmological models with the astronomical observations. Using the Sachs-Wolfe
approximation we calculate spectrum of the super-horizontal CMB anisotropies.
We show that quantum cosmological effects can, in the natural way, explain
suppression of the low CMB multipoles. We show that fine-tuning is not required
here and model is consistent with observations. We also analyse other possible
probes of the quantum cosmologies and discuss perspectives of their
implementation.
