  [Abridged] The measurement of the polarization of the Cosmic Microwave
Background radiation is one of the current frontiers in cosmology. In
particular, the detection of the primordial B-modes, could reveal the presence
of gravitational waves in the early Universe. The detection of such component
is at the moment the most promising technique to probe the inflationary theory
describing the very early evolution of the Universe. We present the updated
performance forecast of the Large Scale Polarization Explorer (LSPE), a program
dedicated to the measurement of the CMB polarization. LSPE is composed of two
instruments: Strip, a radiometer-based telescope on the ground in Tenerife, and
SWIPE (Short-Wavelength Instrument for the Polarization Explorer) a
bolometer-based instrument designed to fly on a winter arctic stratospheric
long-duration balloon. The program is among the few dedicated to observation of
the Northern Hemisphere, while most of the international effort is focused into
ground-based observation in the Southern Hemisphere. Measurements are currently
scheduled in Winter 2021/22 for SWIPE, with a flight duration up to 15 days,
and in Summer 2021 with two years observations for Strip. We describe the main
features of the two instruments, identifying the most critical aspects of the
design, in terms of impact into performance forecast. We estimate the expected
sensitivity of each instrument and propagate their combined observing power to
the sensitivity to cosmological parameters, including the effect of scanning
strategy, component separation, residual foregrounds and partial sky coverage.
We also set requirements on the control of the most critical systematic effects
and describe techniques to mitigate their impact. LSPE can reach a sensitivity
in tensor-to-scalar ratio of $\sigma_r<0.01$, and improve constrains on other
cosmological parameters.
