  Quasiprobabilities are mathematical quantities describing the statistics of
measurement outcomes evaluated at two or more times, which incorporate the
incompatibility of the measurement observables and the state of the measured
quantum system. In this tutorial, we present the definition, interpretation and
properties of the main quasiprobabilities known in the literature. We also
discuss techniques to experimentally access a quasiprobability distribution,
ranging from the weak two-point measurement scheme, to a Ramsey-like
interferometric scheme and procedures assisted by an external detector. Once
defined the fundamental concepts following the standpoint of joint
measurability in quantum mechanics, we illustrate the use of quasiprobabilities
in quantum thermodynamics to describe the quantum statistics of work and heat,
and to explain anomalies in the energy exchanges entailed by a given
thermodynamic transformation. On the one hand, in work protocols, we show how
absorbed energy can be converted to extractable work and vice versa due to
Hamiltonian incompatibility at distinct times. On the other hand, in exchange
processes between two quantum systems initially at different temperatures, we
explain how quantum correlations in their initial state may induce cold-to-hot
energy exchanges, which are unnatural between any pair of equilibrium
non-driven systems. We conclude the tutorial by giving simple examples where
quasiprobabilities are applied to many-body systems: scrambling of quantum
information, sensitivity to local perturbations, quantum work statistics in the
quenched dynamics of models that can be mapped onto systems of free fermions,
for instance the Ising model with a transverse field. Throughout the tutorial,
we meticulously present derivations of essential concepts alongside
straightforward examples, aiming to enhance comprehension and facilitate
learning.
