Can we observe quantum effects in large systems?

A very large number of microstates can produce the same macrostate of a large system.  In some of these microstates many constituents are in the same quantum state and, if the system occupied such a microstate, quantum effects would be observable.  But the probability that the system occupies such a microstate is practically zero.  There are, in general, many more microstates with constituents in different quantum state, for which quantum effects average out

However, elementary particles are indistinguishable, and they are either fermions or bosons.  Microscopic composite particles also can be treated like indistinguishable fermions or bosons.  At low enough temperature, when fewer microstates are available, the  facts that the particles are indistinguishable and that fermions obey the Pauli exclusion principle can alter the probability that certain microstates are occupied enough to make quantum effects observable in large systems.

At high temperature, when many quantum states are available to all particles, all particles approximately obey Boltzmann statistics.
P_Boltzman(E) ∝ exp(-E/(kT)).

At very low temperatures, when the total energy is very low and most particles have to occupy fewer low lying states, bosons are much more likely than distinguishable particles to occupy the ground state.  Bosons obey Bose-Einstein statistics, which significantly differs from Boltzmann statistics at low temperatures.  At low temperatures
P_{Bose-Einstein}(E) ∝ 1/(exp(E/kT) - 1).
Bosons can form Bose-Einstein condensates.  (Example:  Superfluid Helium)
 

Fermions obey Fermi-Dirac statistics.
P_Fermi-Dirac(E) = 1/(exp((E-E_F)/kT) + 1).
Since no two fermions can occupy the same quantum state, some fermions must occupy relatively high energy levels even at very low temperature.  If all the particles occupy the lowest energies allowed by the Pauli exclusion principle, then the energy highest filled energy level is called the Fermi energy E_F.  At all temperatures a quantum state of a fermion gas with an energy equal to the Fermi energy has a probability of 0.5 of being occupied.