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We calculate the electronic structure of AA'AA'...-stacked alternating twistN-layer (tNG) graphene for N = 3, 4, 5, 6, 8, 10, 20 layers and bulkalternating twist (AT) graphite systems where the lattice relaxations aremodeled by means of molecular dynamics simulations. We show that the symmetricAA'AA'... stacking is energetically preferred among all interlayer slidinggeometries for progressively added layers up to N=6. Lattice relaxationsenhance electron-hole asymmetry, and reduce the magic angles with respect tocalculations with fixed tunneling strengths that we quantify from few layers tobulk AT-graphite. Without a perpendicular electric field, the largest magicangle flat-band states locate around the middle following the largesteigenvalue eigenstate in a 1D-chain model of layers, while the densityredistributes to outer layers for smaller magic twist angles corresponding tohigher order effective bilayers in the 1D chain. A perpendicular electric fielddecouples the electronic structure into $N$ Dirac bands with renormalized Fermivelocities with distinct even-odd band splitting behaviors, showing a gap forN=4 while for odd layers a Dirac cone remains between the flat band gaps. Themagic angle error tolerance estimated from density of states maxima expandprogressively from $0.05^{\circ}$ in t2G to up to $0.2^{\circ}$ in AT-graphite,hence allowing a greater flexibility in multilayers. Decoupling of tNG into t2Gwith different interlayer tunneling proportional to the eigenvalues of a 1Dlayers chain allows to map tNG-multilayers bands onto those of periodic bulkAT-graphite's at different $k_z$ values. We also obtain the Landau leveldensity of states in the quantum Hall regime for magnetic fields of up to 50~Tand confirm the presence of nearly flat bands around which we can developsuppressed density of states gap regions by applying an electric field in N > 3systems.

Electronic structure of lattice relaxed alternating twist tNG-multilayer graphene: from few layers to bulk AT-graphite