Question: When is quantum coherence evident at macroscopic scales?
About the article "Quantum mechanics", section "Quantum mechanics and classical physics".

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In quantum mechanics, all objects have wave-like properties (see de Broglie waves).
For instance, in Young's double-slit experiment electrons can be used in the place of light waves.
Each electron's wave-function goes through both slits, and hence has two separate split-beams that contribute to the intensity pattern on a screen.
According to standard wave theory (Fresnel, Huygens) these two contributions give rise to an intensity pattern of bright bands due to constructive interference, interlaced with dark bands due to destructive interference, on a downstream screen.
(Each split-beam, by itself, generates a diffraction pattern with less noticeable, more widely spaced dark and light bands.)
This ability to interfere and diffract is related to coherence (classical or quantum) of the wave.
The association of an electron with a wave is unique to quantum theory.
When the incident beam is represented by a quantum pure state, the split beams downstream of the two slits are represented as a superposition of the pure states representing each split beam.
(This has nothing to do with two particles or Bell's inequalities relevant to an entangled state: a 2-body state, a kind of coherence between two 1-body states.)
The quantum description of imperfectly coherent paths is called a mixed state.
A perfectly coherent state has a density matrix (also called the "statistical operator") that is a projection onto the pure coherent state, while a mixed state is described by a classical probability distribution for the pure states that make up the mixture.
Large-scale (macroscopic) quantum coherence leads to novel phenomena, the so-called macroscopic quantum phenomena.
For instance, the laser, superconductivity and superfluidity are examples of highly coherent quantum systems whose effects are evident at the macroscopic scale.
The macroscopic quantum coherence (Off-Diagonal Long-Range Order, ODLRO) & Onsager (1957), C. N. Yang (1962) for laser light, and superfluidity, is related to first-order (1-body) coherence/ODLRO, while superconductivity is related to second-order coherence/ODLRO.
(For fermions, such as electrons, only even orders of coherence/ODLRO are possible.)
Superfluidity in liquid He4 is related to a partial Bose–Einstein condensate.
Here, the condensate portion is described by a multiply-occupied single-particle state.
Cummings & Johnston (1966)Regarding the occurrence of quantum coherence at a macroscopic level, it is interesting to note that the classical electromagnetic field exhibits macroscopic quantum coherence.
The most obvious example is the carrier signal for radio and TV.
They satisfy Glauber's quantum description of coherence. 
 and classical physics.
Predictions of quantum mechanics have been verified experimentally to an extremely high degree of accuracy.
According to the correspondence principle between classical and quantum mechanics, all objects obey the laws of quantum mechanics, and classical mechanics is just an approximation for large systems of objects (or a statistical quantum mechanics of a large collection of particles).
The laws of classical mechanics thus follow from the laws of quantum mechanics as a statistical average at the limit of large systems or large quantum numbers.
However, chaotic systems do not have good quantum numbers, and quantum chaos studies the relationship between classical and quantum descriptions in these systems.
Quantum coherence is an essential difference between classical and quantum theories as illustrated by the Einstein–Podolsky–Rosen (EPR) paradox — an attempt to disprove quantum mechanics by an appeal to local realism.
Quantum interference involves adding together "probability amplitudes", whereas classical "waves" infer that there is an adding together of "intensities".
For microscopic bodies, the extension of the system is much smaller than the coherence length, which gives rise to long-range entanglement and other nonlocal phenomena characteristic of quantum systems.
Quantum coherence is not typically evident at macroscopic scales, though an exception to this rule may occur at extremely low temperatures (i.e.
approaching absolute zero) at which quantum behavior may manifest itself macroscopically.
This is in accordance with the following observations:
 Many macroscopic properties of a classical system are a direct consequence of the quantum behavior of its parts.
For example, the stability of bulk matter (consisting of atoms and molecules which would quickly collapse under electric forces alone), the rigidity of solids, and the mechanical, thermal, chemical, optical and magnetic properties of matter are all results of the interaction of electric charges under the rules of quantum mechanics.
 While the seemingly "exotic" behavior of matter posited by quantum mechanics and relativity theory become more apparent when dealing with particles of extremely small size or velocities approaching the speed of light, the laws of classical, often considered "Newtonian", physics remain accurate in predicting the behavior of the vast majority of "large" objects (on the order of the size of large molecules or bigger) at velocities much smaller than the velocity of light.
Quantum mechanics is most often used to describe matter on the scale of molecules, atoms, or elementary particles.
However some phenomena, particularly at low temperatures, show quantum behavior on a macroscopic scale.
The best-known examples of macroscopic quantum phenomena are superfluidity and superconductivity; another example is the quantum Hall effect.
Since 2000 there has been extensive experimental work on quantum gases, particularly Bose–Einstein Condensates.
Between 1996 to 2003 four Nobel prizes were given for work related to macroscopic quantum phenomena.
Macroscopic quantum phenomena can be observed in superfluid helium and in superconductors, but also in dilute quantum gases and in laser light.
Although these media are very different, their behavior is very similar as they all show macroscopic quantum behavior.
Quantum phenomena are generally classified as macroscopic when the quantum states are occupied by a large number of particles (typically Avogadro's number) or the quantum states involved are macroscopic in size (up to km size in superconducting wires). 
Gomatam’s primary area of research is in non-relativistic quantum mechanics (QM), which emerged in 1925 with Erwin Schrödinger's derivation of his now famous “wave equation”.
This equation applies, "in principle", to both microscopic regimes (atoms, sub-atomic particles) and to other larger regimes (mesoscopic, macroscopic and cosmological scales).
However, "in practice", quantum mechanics began by applying the wave equation to the micro regime, while treating the macroscopic regime as classical.
Since all our laboratory observations are in the macroscopic regime, logically speaking, he says, we should apply the wave equation first to the macro regime before applying it to the micro regime.
In his view, failure to do so in the history of QM has resulted in the famous “measurement problem” and a host of deep conceptual puzzles that plague microscopic QM even today.
Thus, developing the application of QM to the macro regime, call it Macroscopic Quantum Mechanics (MQM), remains the most pressing, fundamental problem in all of modern physics.
See "Physics for the 21st Century", a new course at the Harvard-Smithsonian Center for Astrophysics, designed by Prof. Christopher Stubbs.
Gomatam is developing his own approach to MQM, which is distinct from the ideas of ‘macroscopic dissipative systems’ and ‘macroscopic quantum coherence’, developed in the early 80s by Anthony James Leggett, a Nobel Laureate in physics.
In general, Leggett's attempt is to "indirectly" observe superposition at the macroscopic level by "extending" current microscopic quantum physics to the macroscopic level.
In contrast, Gomatam is attempting to develop MQM "independent of the application of the Schrodinger equation to the micro regime", in such a manner that quantum superposition can be "directly" observed at the macroscopic level.
This involves introducing a new notion of macroscopic objects as quantum "kinds", instead of classical "objects".
In this regard, he is also developing two further new ideas within physics: the ontology of “Objective, Semantic Information” (OSI) and corresponding “Relational Properties” (RPs).
As part of developing his version of MQM, Gomatam has related interests in exotic manifolds, semantic information processing, quantum computation, and philosophy of ordinary language. 
  A Bose–Einstein condensate (BEC) is a collection of boson atoms that are all in the same quantum state.
In a thermodynamic system, the ground state becomes macroscopically occupied below a critical temperature — roughly when the thermal de Broglie wavelength is longer than the interatomic spacing.
Superfluidity in liquid Helium-4 is believed to be associated with the Bose–Einstein condensation in an ideal gas.
But 4He has strong interactions, and the liquid structure factor (a 2nd-order statistic) plays an important role.
The use of a coherent state to represent the superfluid component of 4He provided a good estimate of the condensate / non-condensate fractions in superfluidity,consistent with results of slow neutron scattering.
Most of the special superfluid properties follow directly from the use of a coherent state to represent the superfluid component — that acts as a macroscopically occupied single-body state with well-defined amplitude and phase over the entire volume.
(The superfluid component of 4He goes from zero at the transition temperature to 100% at absolute zero.
But the condensate fraction is about 6% at absolute zero temperature, T=0K.)
Early in the study of superfluidity, Penrose and Onsager proposed a metric ("order parameter") for superfluidity.
It was represented by a macroscopic factored component (a macroscopic eigenvalue) in the first-order reduced density matrix.
Later, C. N. Yang proposed a more generalized measure of macroscopic quantum coherence, called "Off-Diagonal Long-Range Order" (ODLRO), that included fermion as well as boson systems.
ODLRO exists whenever there is a macroscopically large factored component (eigenvalue) in a reduced density matrix of any order.
Superfluidity corresponds to a large factored component in the first-order reduced density matrix.
(And, all higher order reduced density matrices behave similarly.)
Superconductivity involves a large factored component in the 2nd-order ("Cooper electron-pair") reduced density matrix.
The reduced density matrices used to describe macroscopic quantum coherence in superfluids are formally the same as the correlation functions used to describe orders of coherence in radiation.
Both are examples of macroscopic quantum coherence.
The macroscopically large coherent component, plus noise, in the electromagnetic field, as given by Glauber's description of signal-plus-noise, is formally the same as the macroscopically large superfluid component plus normal fluid component in the two-fluid model of superfluidity.
Every-day electromagnetic radiation, such as radio and TV waves, is also an example of near coherent states (macroscopic quantum coherence).
That should "give one pause" regarding the conventional demarcation between quantum and classical.
The coherence in superfluidity should not be attributed to any subset of helium atoms; it is a kind of collective phenomena in which all the atoms are involved (similar to Cooper-pairing in superconductivity, as indicated in the next section).